Compactly Generated Stacks: a Cartesian Closed Theory Of

Home , Exponential object

arXiv:0907.3925v3 [math.AT] 19 Jan 2012 OPCL EEAE TCS ATSA CLOSED CARTESIAN A STACKS: GENERATED COMPACTLY Introduction . 1 2 tcsfrteCmatyGnrtdGohnic Topology Grothendieck Generated Compactly the for Topology Stacks Grothendieck Generated Compactly .2. Topology The 3 Grothendieck Generated Compactly .1. Stacks The 3 Mapping and Stacks . Paratopological 3 Groupoids Topological .2. Fibrant 2 Stacks .1. Topological Clos 2 Results Cartesian Main Spaces and . Hausdorff Organization 2 Generated Compactly are .2. Why 1 .1. 1 4 oooia Stacks Topological References Groupoids Topological of .4. Bicategories B Bundles .3. Principal B Examples Some and .2. Stacks Definitions B Topological and Groupoids .1. Topological B . Stacks B Stacks Topological Stacks with . Generated Comparison A Compactly of Stacks Types .4. Generated Homotopy 4 Compactly of Stacks .3. Mapping 4 Stacks Generated .2. Compactly 4 Stacks Generated .1. Compactly 4 . 4 hnte etitt h aesakoe oal opc Hau compact locally over equivalent. topol stack homotopy same same are the the to by topolo restrict presentation classical they a a then have If stack case. generated classical pactly the gr in topological as Compa of just bicategory bundles, atlases. a to Hausdorff equivalent also compact are stacks locally cl those admit and which a stacks is stacks there generated fact, compactly classical In between of topology. categories Grothendieck analogue different the a cal for is bicategory, but stacks, This generated closed. compactly Cartesian of and complete both is Abstract. HOYO OOOIA STACKS TOPOLOGICAL OF THEORY ovnetbctgr ftplgclsak sconstruct is stacks topological of bicategory convenient A AI CARCHEDI DAVID Contents 1 ia tc n com- a and stack gical uod n principal and oupoids qiaec fbi- of equivalence n oooia stacks, topological e h bicategory the led sia topological assical dr pcsand spaces sdorff gclgroupoid, ogical tygenerated ctly dwhich ed ed? 50 48 46 44 43 43 41 32 27 25 21 21 20 14 13 9 8 8 5 4 2 2 David Carchedi

1 . Introduction The aim of this paper is to introduce the bicategory of compactly generated stacks. Compactly generated stacks are “essentially the same” as topological stacks, however, their associated bicategory is Cartesian closed and complete, whereas the bicategory of topological stacks appears to enjoy neither of these properties. In this paper, we show that these categorical shortcomings can be overcame by refining the open cover Grothendieck topology to take into account compact generation. It is well known that the category of topological spaces is not well behaved. In particular, it is not Cartesian closed. Recall that if a category C is Cartesian closed then for any two objects X and Y of C , there exists a mapping object Map (X, Y ) , such that for every object Z of C , there is a natural isomorphism Hom (X, Map (X, Y )) =∼ Hom (Z × X, Y ) . The category of topological spaces is not Cartesian closed; one can topologize the set of maps from X to Y with the compact-open topology, but this space will not always satisfy the above universal property. In a 1967 paper [24], Norman Steenrod set forth compactly generated Hausdorff spaces as a convenient category of topological spaces in which to work. In particular, compactly generated spaces are Cartesian closed. Though technical in nature, history showed this paper to be of great importance; it is now standard practice to work within the framework of compactly generated spaces. Unfortunately, topological stacks are not as nicely behaved as topological spaces, even when considering only those associated to compactly generated Hausdorff topological groupoids. The bicategory of topological stacks is deficient in two ways as it appears to be neither complete, nor Cartesian closed [20],[21]; that is, mapping stacks need not exist. Analogously to the definition of mapping spaces, if X and Y are two topological stacks, a mapping stack Map (X , Y ) (if it exists), would be a topological stack such that there is a natural equivalence of groupoids Hom (Z , Map (X , Y )) ≃ Hom (Z × X , Y ) , for every topological stack Z . The mapping stack Map (X , Y ) always exists as an abstract stack, but it may not be a topological stack, so we may not be able to apply all the tools of topology to it. This problem can be fixed however, as there exists a more well behaved bicategory of topological stacks, which we call “compactly generated stacks”, which is Cartesian closed and complete as a bicategory. This bicategory provides the topologist with a convenient bicategory of topological stacks in which to work. The aim of this paper is to introduce this theory. The study of mapping stacks has been done in many different settings. In the algebraic world, Masao Aoki and Martin Olsson studied the existence of mapping stacks between algebraic stacks in [2], and [22] respectively. The special case of differentiable maps between orbifolds has been studied by Weimin Chen in [4], and is restricted to the case where the domain orbifold is compact. AndréHaefliger has studied the case of smooth maps between étale Lie groupoids (which correspond to differentiable stacks with an étale atlas) in [8]. In [9], Ernesto Lupercio and Bernardo Uribe showed that the free loop stack (the stack of maps from S1 to the stack) of an arbitrary topological stack is again a topological stack. In [21], Behrang Noohi addressed the general case of maps between topological stacks. He showed that under a certain compactness condition on the domain stack, the stack of Compactly Generated Stacks 3 maps between two topological stacks is a topological stack, and if this compactness condition is replaced with a local compactness condition, the mapping stack is “not very far” from being topological. In order to obtain a Cartesian closed bicategory of topological stacks, we first restrict to stacks over a Cartesian closed subcategory of the category TOP of all topological spaces. For instance, all of the results of [21] about mapping stacks are about stacks over the category of compactly generated spaces with respect to the open cover Grothendieck topology. We choose to work over the category of compactly generated Hausdorff spaces (CGH) since, in addition to being Cartesian closed, every compact Hausdorff space is locally compact Hausdorff, which is crucial in defining the compactly generated Grothendieck topology. There are several equivalent ways of describing compactly generated stacks. The description that substantiates most clearly the name “compactly generated” is the description in terms of topological groupoids and principal bundles. Recall that the bicategory of topological stacks is equivalent to the bicategory of topological groupoids and principal bundles. Classically, if X is a topological space and G is a topological groupoid, the map π of a (left) principal G-bundle P over X

G1 P ⑧ µ ⑧ ⑧⑧ π ⑧⑧ ⑧⑧ G0 X must admit local sections. If instead π only admits local sections over each compact subset of X, then one arrives at the definition of a compactly generated principal bundle. With this notion of compactly generated principal bundles, one can define a bicategory of topological groupoids in an obvious way. This bicategory is equivalent to compactly generated stacks. There is another simple way of defining compactly generated stacks. Given any stack X over the category of compactly generated Hausdorff spaces, it can be restricted to the category of compact Hausdorff spaces CH. This produces a 2-functor j∗ : TSt → St (CH) from the bicategory of topological stacks to the bicategory of stacks over compact Hausdorff spaces. Compactly generated stacks are (equivalent to) the essential image of this 2-functor. Finally, the simplest description of compactly generated stacks is that compactly generated stacks are classical topological stacks (over compactly generated Haus- dorff spaces) which admit a locally compact Hausdorff atlas. In this description, the mapping stack of two spaces is usually not a space, but a stack! For technical reasons, neither of the three previous concepts of compactly generated stacks are put forth as the definition. Instead, a Grothendieck topology CG is introduced on the category CGH of compactly generated Hausdorff spaces which takes into account the compact generation of this category. It is in fact the Grothendieck topology induced by geometrically embedding the topos Sh (CH) CGHop of sheaves over CH into the topos of presheaves Set . Compactly generated stacks are defined to be presentable stacks (see Definition B .17) with respect to this Grothendieck topology. The equivalence of all four notions of compactly generated stacks is shown in Section 4 .1. 4 David Carchedi

1 .1. Why are Compactly Generated Hausdorff Spaces Cartesian Closed? In order to obtain a Cartesian closed bicategory of topological stacks, we start with a Cartesian closed category of topological spaces. We choose to work with the aforementioned category CGH of compactly generated Hausdorff spaces (also known as Kelley spaces). Definition 1 .1. A topological space X is compactly generated if it has the final topology with respect to all maps into X with compact Hausdorff domain. When X is Hausdorff, this is equivalent to saying that a subset A of X is open if and only if its intersection which every compact subset of X is open. -The inclusion v : CGH ֒→ HAUS of the category of compactly generated Haus dorff spaces into the category of Hausdorff spaces admits a right adjoint k, called the Kelley functor, which replaces the topology of a space X with the final topology with respect to all maps into X with compact Hausdorff domain. Limits in CGH are computed by first computing the limit in HAUS, and then applying the Kelley functor. (In this way the compactly generated product topology differs from the ordinary product topology). In particular, CGH is a complete category. Although the fact that this category is Cartesian closed is a classical result (see: [24]), we will recall briefly the key reasons why this is true in order to gain insight into how one could construct a Cartesian closed theory of topological stacks. (1) In TOP, if K is compact Hausdorff, then for any space X, the space of maps endowed with the compact-open topology serves as an exponential object Map (K,X). (2) A Hausdorff space Y is compactly generated if and only if it is the colimit of all its compact subsets: Y = lim K . −→ α Kα֒→Y

i) For a ﬁxed Y , for all X, Map (Kα,X) exists for each compact subset Kα of Y . ii) CGH has all limits So by general properties of limits and colimits, the space

Map (Y,X) := lim Map (K ,X) 1 ←− α Kα֒→Y is a well deﬁned exponential object (with the correct universal property). The story starts the same for topological stacks:

Let Y be as above and let X be a topological stack. Then Map (Kα, X ) is a topological stack for each compact subset Kα ⊂ Y (see [21]). One might therefore be tempted to claim:

Map (Y, X ) := holim Map (Kα, X ) , ←−−−−−−− Kα֒→Y but there are some problems with this. First of all, this weak limit may not exist as a topological stack, since topological stacks are only known to have ﬁnite weak

1Technically, some of the spaces Map (Kα,X) may be fail to be compactly generated, so we must actually take the limit after applying the Kelley functor to each of these mapping spaces. Compactly Generated Stacks 5 limits. There is also a more technical problem related to the fact that the Yoneda embedding does not preserve colimits (see Section 3 for details). The main task of this paper is to show that both of these difficulties can be surmounted by using a more suitable choice of Grothendieck topology on the category of compactly generated Hausdorff spaces. The resulting bicategory of presentable stacks with respect to this topology will be the bicategory of compactly generated stacks and turn out to be both Cartesian closed and complete. 1 .2. Organization and Main Results. Section 2 is a review of some recent developments in topological groupoids and topological stacks, including some results of David Gepner and AndréHenriques in [7] which are crucial for the proof of the completeness and Cartesian closedness of compactly generated stacks. In this section, we also extend Behrang Noohi’s results to show that the mapping stack of two topological stacks is topological if the domain stack admits a compact Haus- dorff atlas and “nearly topological” if the domain stack admits a locally compact Hausdorff atlas. Section 3 details the construction of the compactly generated Grothendieck topology CG on the category of compactly generated Hausdorff spaces CGH. This is the Grothendieck topology whose associated presentable stacks are precisely compactly generated stacks. Many properties of the associated categories of sheaves and stacks are derived. Section 4 is dedicated to compactly generated stacks. In Section 4 .1, it is shown that compactly generated stacks are equivalent to two bicategories of topological groupoids. Also, it is shown that these are in turn equivalent to the restriction of topological stacks to compact Hausdorff spaces. Finally, it is shown that compactly generated stacks are equivalent to ordinary topological stacks (over compactly generated Hausdorff spaces) which admit a locally compact Hausdorff atlas. Section 4 .1 also contains one of the main results of the paper: Theorem 1 .1. The bicategory of compactly generated stacks is closed under arbitrary small weak limits. (See Corollary 4 .3). Section 4 .2 is dedicated to the proof of the main result of the paper: Theorem 1 .2. If X and Y are arbitrary compactly generated stacks, then Map (Y , X ) is a compactly generated stack. (See Theorem 4 .8). This of course proves that classical topological stacks (over compactly generated Hausdorff spaces) which admit a locally compact Hausdorff atlas form a Cartesian closed and complete bicategory. We also give a concrete description of a topological groupoid presentation for the mapping stack of two compactly generated stacks. Section 4 .3 uses techniques developed in [20] to assign to each compactly generated stack a weak homotopy type. Finally, in section 4 .4, there is a series of results showing how compactly generated stacks are “essentially the same” as topological stacks. In particular we extend the construction of a weak homotopy type to a wider class of stacks which include all topological stacks and all compactly generated stacks, so that their corresponding homotopy types can be compared. For instance, the following theorems are proven: 6 David Carchedi

Theorem 1 .3. For every topological stack X , there is a canonical compactly generated stack X¯ and a map X → X¯ which induces an equivalence of groupoids X (Y ) → X¯ (Y ) for all locally compact Hausdorff spaces Y . Moreover, this map is a weak homotopy equivalence. (See Corollary 4 .10). Theorem 1 .4. Let X and Y be topological stacks such that Y admits a locally compact Hausdorff atlas. Then Map (Y , X ) is “nearly topological”, Map Y¯, X¯ is a compactly generated stack, and there is a canonical weak homotopy equivalence Map (Y , X ) → Map Y¯, X¯ . Moreover, Map (Y , X ) and Map Y¯, X¯ restrict to the same stack over locally compact Hausdorff spaces. (See Theorem 4 .19). We end this paper by showing in what way compactly generated stacks are to topological stacks what compactly generated spaces are to topological spaces: Recall that there is an adjunction

k CG o TOP, v / exhibiting compactly generated spaces as a co-reﬂective subcategory of the category of topological spaces, and for any space X, the co-reﬂector vk (X) → X is a weak homotopy equivalence. We prove the 2-categorical analogue of this statement: Theorem 1 .5. There is a 2-adjunction

k CG TSt o TSt, v /

v ⊥ k, exhibiting compactly generated stacks as a co-reflective sub-bicategory of topological stacks, and for any topological stack X , the co-reflector vk (X ) → X is a weak homotopy equivalence. A topological stack is in the essential image of the 2-functor v : CG TSt → TSt if and only if it admits a locally compact Hausdorff atlas. (See Theorem 4 .20.)

Acknowledgment: This paper was written during my studies as a Ph.D. student at Utrecht University. I would like to thank my adviser Ieke Moerdijk for his guidance and patience; it was due to him that I came to consider the question Compactly Generated Stacks 7 of the presentability of the mapping stack of two topological stacks, starting first through the language of topological groupoids and étendues. I would also like to thank Marius Crainic for sparking my interest, in the first place, in topological groupoids and Lie groupoids in the form of a very interesting seminar as part of an MRI masterclass. I would very much like to thank AndréHenriques as well for countless hours of useful mathematical discussions and suggestions. I would like to thank AndréHaefliger for taking the time to explain to me his work on maps between étale Lie groupoids during a visit to Genève. I would also like to thank Thomas Goodwillie for help with a homotopy-theoretic proof, and Behrang Noohi for his helpful e-mail correspondence. Finally, I would like to thank the referees of this paper whose comments have been quite helpful. 8 David Carchedi

2 . Topological Stacks A review of the basics of topological groupoids and topological stacks including many notational conventions used in this section can be found in Appendix B .4. Remark. In this paper, we will denote the bicategory of topological stacks by TSt. 2 .1. Fibrant Topological Groupoids. The notion of fibrant topological groupoids was introduced in [7]. Roughly speaking, fibrant topological groupoids are topological groupoids which “in the eyes of paracompact Hausdorff spaces are stacks.” The fact that every topological groupoid is Morita equivalent to a fibrant one is essential to the existence of arbitrary weak limits of compactly generated stacks. Since this concept is relatively new, in this subsection, we summarize the basic facts about fibrant topological groupoids. All details may be found in [7]. Definition 2 .1. The classifying space of a topological groupoid G is the fat geometric realization of its simplicial nerve (regarded as a simplicial space) and is denoted by kGk. For any topological groupoid G, the classifying space of its translation groupoid kEGk (see Definition B .8) admits the structure of a principal G-bundle over the classifying space kGk. Definition 2 .2. [7] Let G be a topological groupoid. A principal G-bundle E over a space B is universal if every principal G-bundle P over a paracompact Hausdorff base X admits a G-bundle map

P / E

X / B, unique up to homotopy. Lemma 2 .1. [7] The principal G-bundle

G1 kEGk µ ⑤⑤ ⑤⑤ π ⑤⑤ }⑤⑤ G0 kGk is universal. Definition 2 .3. [7] A topological groupoid G is fibrant if the unit principal G-bundle is universal. (See Definition B .10.) Definition 2 .4. [7] The fibrant replacement of a topological groupoid G is the gauge groupoid of the universal principal G-bundle kEGk, denoted Fib (G). (See Definition B .13.) Remark. If G is compactly generated Hausdorff, then so is Fib (G). Lemma 2 .2. [7] Fib (G) is fibrant. Compactly Generated Stacks 9

Lemma 2 .3. [7] There is a canonical groupoid homomorphism

ξG : G → Fib (G) which is a Morita equivalence for all topological groupoids G. The following theorem will be of importance later: Theorem 2 .4. [7] Let G be a ﬁbrant topological groupoid. Then for any topological groupoid H with paracompact Hausdorﬀ object space H0, there is an equivalence of groupoids

HomTSt ([H] , [G]) ≃ HomTOPGpd (H, G) natural in H. In particular, the restriction of y˜(G) to the full subcategory of paracompact Hausdorff spaces agrees with the restriction of [G]. (See Appendix B .4 for the notation.) 2 .2. Paratopological Stacks and Mapping Stacks. Stacks on TOP (with respect to the open cover topology) come in many different flavors. Of particular importance of course are topological stacks, which are those stacks coming from topological groupoids. However, this class of stacks seems to be too restrictive since many natural stacks, for instance the stack of maps between two topological stacks, appear to not be topological. A topological stack is a stack X which admits a representable epimorphism X → X from a topological space X. This implies: i) Any map T → X from a topological space is representable (equivalently, the diagonal ∆ : X → X × X is representable) [19]. ii) If T → X is any map, then the induced map T ×X X → T admits local sections (i.e. is an epimorphism in TOP). If the second condition is slightly weakened, the result is a stack which is “nearly topological”. Definition 2 .5. [20] A paratopological stack is a stack X on TOP (with respect to the open cover topology), satisfying condition i) above, and satisfying condition ii) for all maps T → X from a paracompact Hausdorff space2. Paratopological stacks are very nearly topological stacks: Proposition 2 .1. [20] A stack X with representable diagonal is paratopological if and only if there exists a topological stack X¯ and a morphism q : X¯ → X such that for any paracompact Hausdorff space T, q induces an equivalence of groupoids

(1) q (T ): X¯ (T ) → X (T ) . The idea of the proof can be found in [20], but is enlightening, so we include it for completeness: If q is as in (1), and p : X → X¯ is an atlas for X¯, then q ◦ p : X → X

2In [20], this Hausdorff condition does not explicitly appear, but it is necessary for the theorems proven in [20] and [21]. 10 David Carchedi satisfies condition ii) of Definition 2 .5, hence X is paratopological. Conversely, if X is paratopological, take p : X → X as in condition ii) of Definition 2 .5. Form the weak fibered product

X ×X X / X

p p X / X . Let X¯ be the topological stack associated with the topological groupoid

X ×X X ⇒ X, and q : X¯ → X the canonical map. Deﬁnition 2 .6. Any two stacks X and Y on TOP have an exponential stack X Y such that

X Y (T ) = Hom (Y × T, X ) . We will from here on in denote X Y by Map (Y , X ) and refer to it as the mapping stack from Y to X . For the rest of this section, we work in the category CGH of compactly generated Hausdorﬀ spaces, which is Cartesian closed. In [21] Noohi proved:

Theorem 2 .5. If X and Y are topological stacks with Y ≃ [H] with H0 and H1 compact Hausdorﬀ3, then Map (Y , X ) is a topological stack. It appears that when Y does not satisfy this rather rigid compactness condition, that this may fail (however, we will shortly release the condition for the arrow space). Noohi also proved that when Y is instead locally compact, then Map (Y , X ) is at least paratopological: Theorem 2 .6. [21] If X and Y are paratopological stacks with Y ≃ [H] such that H0 and H1 are locally compact, then Map (Y , X ) is a paratopological stack. We end this section by extending Noohi’s results to work without imposing conditions on the arrow space. Firstly, Noohi’s proof of Theorem 2 .5 easily extends to the case when H1 need not be compact: First, we will need a lemma from [21]: Lemma 2 .7. Let X and Y be topological stacks. Then the diagonal of the stack Map (Y , X ) is representable, i.e., every morphism T → Map (Y , X ) , with T a topological space, is representable. Theorem 2 .8. If X and Y are topological stacks and Y admits a compact Hausdorﬀ atlas, then Map (Y , X ) is a topological stack.

3Noohi claimed that this works without assuming the Hausdorff condition and working with compactly generated spaces (the monocoreflective hull of CH in TOP), however, his proof seems to have a gap without this Hausdorff assumption; within it, it is (implicitly) assumed that the product of a (non-Hausdorff) compact space with a compactly generated space is compactly generated, but a non-Hausdorff compact space is not necessarily locally compact (in the sense that each point has a neighborhood base of compact sets). Compactly Generated Stacks 11

Proof. Fix a compact Hausdorff atlas K → Y for Y . Let K denote the corresponding topological groupoid K ×Y K ⇒ K. Let U denote the set of finite open covers of K. For each U ∈U, consider the Cechˇ KU groupoid KU (See Definition B .5) and G , the internal exponent of groupoid objects in compactly generated Hausdorff spaces. KU Set RU := G 0 . By adjunction, the canonical map

KU RU → G induced by the unit, produces a homomorphism

RU × KU → G.

Suppose U is given by U = (Ni) , and let V be the open cover of RU × K, given by

(RU × Ni) . Then this homomorphism is a map

(RU × K)V → G, and in particular, a generalized homomorphism from RU × K to G (See Deﬁnition B .16). This corresponds to a map of stacks

RU × Y → X , which by adjunction is a morphism

pU : RU → Map (Y , X ) . Let R := RU . U J a∈ Then we can conglomerate these morphisms to a morphism p : R → Map (Y , X ) . We will show that p is an epimorphism. By Lemma 2 .7, it is representable. Suppose that f : T → Map (Y , X ) is any morphism from a space T . We will show that f locally factors through p up to isomorphism. By adjunction, f corresponds to a map f¯ : T × Y → X , which corresponds to a generalized homomorphism ˜ f : (T × K)W → G for some open cover W of T × K. Let t ∈ T be an arbitrary point. Since K is compact Hausdorﬀ, T ×K, with the classical product topology, is already compactly generated, so we can assume, without loss of generality, that each element Wj of the cover W is of the form Vj × Uj for open subsets Vj and Uj of T and K respectively. Let t ∈ T be an arbitrary point. Then as W covers the slice {t}× K, which is compact, there exists a ﬁnite collection (U × V ) which covers it. Let j j j∈At

Ot := ∩ Uj. j∈At 12 David Carchedi

Then Ot is a neighborhood of t in T such that for all j ∈ At,

Ot × Vj ⊂ Wj . Let U = (V ) be the corresponding ﬁnite cover of K, and t j j∈At N = (O × V ) t t j j∈At the cover of Ot × K. Denote the composite f˜ O × K ∼ (O × K) → (T × K) t Ut = t Nt W −→ G by f˜t. By adjunction, this corresponds to a homomorphism

KU Ot → G t , so the induced map of stacks

Ot → Map (Y , X ) factors through p. This induced map of stacks is the same as the map adjoint to the one induced by f˜t, f¯ Ot × Y → T × Y −→ X , so we are done. Now, we will recall some basic notions from topology:

Definition 2 .7. A shrinking of an open cover (Uα)α∈A is another open cover (Vα)α∈A indexed by the same set such that for each α, the closure of Vα is contained in Uα. A topological space X is a shrinking space if and only if every open cover of X admits a shrinking. The following proposition is standard: Proposition 2 .2. Every paracompact Hausdorff space is a shrinking space. Theorem 2 .9. If X and Y are topological stacks such that Y admits a locally compact Hausdorff atlas, then Map (Y , X ) is paratopological. Proof. It is proven in [21] that if X and Y are topological stacks, then Map (Y , X ) has a representable diagonal. Therefore, by Proposition 2 .1, it suffices to prove that there exists a topological stack Z and a map q : Z → Map (Y , X ) which induces an equivalence of groupoids q (T ): Z (T ) → Map (Y , X ) (T ) along every paracompact Hausdorff space T . Let G be a topological groupoid presenting X . Let Y → Y be a locally compact Hausdorff atlas for Y . Then we may find a covering of Y by compact neighborhoods, (Yα) . It follows that Yα → Y → Y α is an atlas. Hence we can choosea a topological groupoid H presenting Y such H that H0 is a disjoint union of compact Hausdorff spaces. Let Fib (G) denote the internal exponent of groupoid objects in compactly generated Hausdorff spaces. Let K := Fib (G)H . h i Compactly Generated Stacks 13

Note that there is a canonical map ϕ : K → Map (Y , X ) H which sends any generalized homomorphism TU → Fib (G) to the induced generalized homomorphism from T × H, TU × H → Fib (G) (which may be viewed as object in HomTSt (T × Y , X ) since G and Fib (G) are Morita equivalent). Suppose that T is a paracompact Hausdorﬀ space. Then:

H K (T ) ≃ holim HomCGHGpd TU ,Fib (G) −−−−−−−→ U ≃ holim HomCGHGpd (TU × H,Fib (G)) . −−−−−−−→ U

Note that any paracompact Hausdorff space is a shrinking space so without loss of generality we may assume that each cover U of T is a topological covering by closed neighborhoods. Since any closed subset of a paracompact space is paracompact, this means that the groupoid TU has paracompact Hausdorff object space. Moreover, the object space of TU × H is the product of the compactly generated space TU and the locally compact Hausdorff space H0, hence compactly generated. Since TU has paracompact Hausdorff object space, and H0 is a disjoint union of compact Hausdorff spaces, the product is in fact paracompact Hausdorff by [16]. Finally, by Theorem 2 .4, we have that

HomCGGpd (TU × H,Fib (G)) ≃ HomTSt (T × Y , X ) . Hence K (T ) ≃ Map (Y , X ) (T ) .

3 . The Compactly Generated Grothendieck Topology Recall from section 1 .1 that if Y is a compactly generated Hausdorﬀ space and X a topological stack, then one might be tempted to claim that

Map (Y, X ) := holim Map (Kα, X ) , ←−−−−−−− Kα֒→Y where the weak limit is taken over all compact subsets Kα of Y . However, there are some immediate problems with this temptation: • This weak limit may not exist as a topological stack. • The fact that Y is the colimit of its compact subsets in CGH does not imply that Y is the weak colimit of its compact subsets as a topological stack since the Yoneda embedding does not preserve arbitrary colimits. Recall however that for an arbitrary subcanonical Grothendieck site (C ,J), the Yoneda embedding y : C ֒→ ShJ (C ) preserves colimits of the form C = lim C −→ α Cα→C

fα where Cα → C is a J-cover. We therefore shall construct a Grothendieck topology CG on CGH, called the compactly generated Grothendieck topology, such that for all Y , the inclusion of all compact subsets (Kα ֒→ Y ) is a CG -cover. As it shall 14 David Carchedi turn out, in addition to being Cartesian closed, the bicategory of presentable stacks for this Grothendieck topology will also be complete. In this subsection, we give a geometric construction of the compactly generated Grothendieck topology on CGH. Those readers not familiar with topos theory may wish to skip to Definition 3 .1 for the concrete definition of a CG -cover. Some important properties of CG -covers are summarized as follows: i) Every open cover is a CG -cover (Proposition 3 .2) ii) For any space, the inclusion of all its compact subsets is a CG -cover (Corol- lary 3 .1) iii) Every CG -cover of a locally compact Hausdorff space can be refined by an open one (Proposition 3 .4) iv) The category of CG -sheaves over compactly generated Hausdorff spaces is equivalent to the category of ordinary sheaves over compact Hausdorff spaces (Theorem 3 .2). 3 .1. The Compactly Generated Grothendieck Topology. Let j : CH ֒→ CGH be the full and faithful inclusion of compact Hausdorff spaces into compactly gen- ∗ erated Hausdorff spaces. This induces a geometric morphism (j , j∗)

∗ CHop j CGHop Set o / Set j∗ 4 which is an embedding (i.e. j∗ is full and faithful [11]) . Denote by

CHop yCH : CGH → Set the functor which assigns a compactly generated Hausdorﬀ space X the presheaf T 7→ HomCGH (T,X) and by

CGHop yCGH : CH → Set the functor which assigns a T ∈ CH the presheaf X 7→ HomCGH (X,T ). Note that ∗ yCH is a fully faithful embedding. The pair (j , j∗) can be constructed as the adjoint pair induced by left Kan extending yCH along the Yoneda embedding. Explicitly:

j∗ (F ) (T )= F (T ) and op j∗ (G) (X) = HomSetCH (yCH (X) , G) . From the general theory of adjoint functors, j∗ restricts to an equivalence be- CHop tween, on one hand the full subcategory of Set whose objects are those for which the co-unit ε (j) is an isomorphism, and on the other hand, the full subcategory of CGHop Set whose objects are those for which the unit η (j) is an isomorphism. How- ever, since j is fully faithful, the co-unit is always an isomorphism, which veriﬁes that j∗ is fully faithful and gives us a way of describing its essential image.

4Technically, these categories are not well deﬁned due to set-theoretic issues, however, this can be overcame by careful use of Grothendieck universes. We will not dwell on this and all such similar size issues in this paper. Compactly Generated Stacks 15

∗ ∗ In fact, j also has a left adjoint j!. The adjoint pair j! ⊥ j is the one induced by yCGH. Hence, j! is the left Kan extension of yCGH. We conclude that j! is also fully faithful (see: [11]). Denote by Sh (CGH) the topos of sheaves on compactly generated Hausdorff spaces with respect to the open cover topology. We define Sh (CH) as the unique topos fitting in the following pullback diagram:

CH CHop Sh ( ) / / Set

CGH op Sh (CGH) / / Set Due to the factorization theorem of geometric morphisms in topos theory [11], the CHop geometric embedding Sh (CH) ֌ Set corresponds to a Grothendieck topology K on CH. It is easy to verify that since the functor j is fully faithful, the covering sieves in K for a compact Hausdorff space K are precisely those subobjects S ֌ y (K) which are obtained by restricting a covering sieve of K with respect to the open cover topology on CGH to CH via the functor j∗. In this sense, the covering sieves are “the same as in the open cover topology”. Proposition 3 .1. The Grothendieck topology K on CH has a basis of finite n covers of the form (Ti ֒→ T )i=1 by compact neighborhoods (i.e. their interiors form an open cover). Proof. Compact Hausdorff spaces are locally compact in the strong sense that every point has a local base of compact neighborhoods. Hence covers by compact neighborhoods generate the same sieves as open covers. Consider the geometric embedding

a CHop Sh (CH) o / Set , i where a denotes the sheafification with respect to K . CGHop Remark. It is clear that for any presheaf F in Set , the K -sheafification of the restriction of F to CH is the same as the restriction of the sheafification of F. By composition, we get an embedding of topoi

∗ a◦j CGHop Sh (CH) o / Set . j∗◦i Again by the factorization theorem [11], there exists a unique Grothendieck topology CG on CGH such that the category of sheaves ShCG (CGH) is j∗ (Sh (CH)). We will construct it and give some of its properties. There is a very general construction [11] that shows how to extract the unique Grothendieck topology corresponding to this embedding. CGHop First, we deﬁne a universal closure operation on Set . (For details, see [11] V.1 on Lawvere-Tierney topologies.) Let F be a presheaf over CGH and let 16 David Carchedi m : A ֌ F be a representative for a subobject A of F . Then a representative for the subobject A¯ is given by the left hand side of the following pullback diagram

∗ A¯ / j∗aj (A)

∗ j∗aj (m) ηF ∗ F / j∗aj (F ) , ∗ where η is the unit of the adjunction a ◦ j ⊥ j∗ ◦ i. To describe the covering sieves of CG , it suﬃces to describe the universal closure operation on representables. Let X be a compactly generated Hausdorﬀ space.

Claim. The unit ηX is an isomorphism. Proof. The restriction j∗X is a K -sheaf. Hence ∗ j∗aj X =∼ j∗yCH (X) . Furthermore, for any compactly generated Hausdorﬀ space Y ,

op ∼ j∗yCH (X) (Y ) = HomSetCH (yCH (Y ) ,yCH (X)) = HomCGH (Y,X) since yCH is fully faithful.

Now, let m : A ֌ X a sieve. Then, since the unit ηX is an isomorphism, A¯ is represented by the monomorphism ∗ j∗aj (A) ֌ X. The covering sieves in CG of X are exactly those sieves on X whose closure is equal to the maximal sieve, i.e. X. So m : A ֌ X is a covering sieve if and only if ∗ ∗ m¯ : j∗aj (A) → j∗j (X) =∼ X is an isomorphism. Since j∗ is fully faithful, this is if and only if m˜ : aj∗ (A) → j∗X is an isomorphism. In other words, m : A ֌ X is a covering sieve if and only if the K -sheafification of j∗ (A) is isomorphic to yCH (X) Definition 3 .1. Let X be a compactly generated Hausdorff space and let

αi : Vi ֒→ X)i∈I) be family of inclusions of subsets Vi of X. Such a family is called a CG -cover if for any compact subset K of X, there exists a (ﬁnite) subset J (K) ⊆ I such that the collection (Vj ∩ K)j∈J(K) can by reﬁned by an open cover of K. Denote the set of CG -covers of X by B (X) . Lemma 3 .1. B is a basis for the Grothendieck topology CG . Compactly Generated Stacks 17

Proof. Let (fi : Qi → X) be a class of maps into X. We denote the sieve it generates by Sf . For any compactly generated Hausdorﬀ space Y , we have

Sf (Y )= {h : Y → X such that h factors through fi for some i} .

So, Sf is a covering sieve if and only if when restricted to CH, its K -sheafification ∗ is isomorphic to yCH (X). We first note that j (Sf ) is clearly a K -separated ∗ + presheaf. Hence, its sheafification is the same as j (Sf ) . Since yCH (X) is a sheaf, the canonical map ∗ j (Sf ) ֌ yCH (X) factors uniquely as

∗ j (Sf ) / / yCH (X) . s9 ss ss ss s9 s ∗ + j (Sf )

∗ + It suﬃces to see when the map j (Sf ) ֌ yCH (X) is an epimorphism. Let S˜f be the presheaf on CH

˜ n n Sf (T )= U = (Ui)j=1 , (aj ∈ Sf (Uj ))j=1 | ai|Uij = aj |Uij for all i,j .

n∗ + o Then the map j (Sf ) ֌ yCH (X) ﬁts in a diagram

S˜f / yCH (X) . tt9 ttt ttt 9tt ∗ + j (Sf )

It suffices to see when the canonical map S˜f → yCH (X) is point-wise surjective. This is precisely when for any map h : K → X from a compact Hausdorff space K ∈ CH , there exists an open cover (Uj )j of K such that for all j, h|Uj factors through fi for some i. Classes of maps with codomain X with this property constitute a large basis for the Grothendieck topology CG . It is in fact maximal in the sense that S is a covering sieve if and only if it is one generated by a large cover of this form. We will now show that any such large covering family has a refinement by one of the form of the lemma. Let (fi : Qi → X) denote such a (possibly large) family and let

(iα : Kα ֒→ X) denote the inclusion of all compact subsets of X. Then for each α, there exists a α α ﬁnite open cover of Kα, Oj , such that the inclusion of each Oj into X factors through some fj,α : Qj,α → X via a map α α λj : Oj → Qj,α. 18 David Carchedi

α CH Let U := Oj ֒→ X j,α. Let g : L → X be a map with L ∈ . Then g (L)= Kα for some α. Let g −1 α VL := g Oj . g Then VL is an open cover of L such that the restriction of g to any element of the α cover factors through the inclusion of some Oj into X. Hence the sieve generated by U is a covering sieve for CG which refines the sieve generated by (fi : Qi → X). We have the following obvious proposition whose converse is not true: Proposition 3 .2. Any open cover of a space is also a CG cover. In particular, one cover that is quite useful is the following. Corollary 3 .1. For any compactly generated Hausdorff space, the inclusion of all compact subsets is a CG -cover. However, for the category LCH of locally compact Hausdorff spaces, it suffices to work with open covers: Proposition 3 .3. Every CG -cover of a locally compact Hausdorff space X ∈ LCH can be refined by an open covering. LCH CG (Proof. Let X ∈ and let V = (αi : Vi ֒→ X)i∈I be a -cover of X. Let (Kl be a topological covering of X by compact subsets such that the interiors int (Kl) constitute an open cover for X. Then for each Kl, there exists a finite subset J (Kl) ⊆ I such that

(V ∩ K ) j l j∈J(Kl) can be reﬁned by an open cover (Wj ) for Kl such that the inclusion of each j∈J(Kl) Wj into X factors through the inclusion of Vj . Let U := (Wj ) . Then U is l,j∈J(Kl) an open cover of X reﬁning V.

We can now define the CG -sheafification functor aCG either by the covering sieves, or by using the basis B (i.e. both will give naturally isomorphic functors). Let ShCG (CGH) denote the category of CG -sheaves. Then we have an embedding of topoi given by

aCG CGHop ShCG (CGH) o / Set , ℓ CGHop .where ℓ : ShCG (CGH) ֒→ Set is the inclusion of the category of sheaves By the previous observation that open covers are CG -covers, we also have

ShCG (CGH) ⊂ Sh (CGH) , where Sh (CGH) is the category of sheaves on CGH with respect to the open cover topology. By construction, we have the following theorem: Theorem 3 .2. There is an equivalence of topoi

a◦j∗◦ℓ Sh (CH) o / ShCG (CGH) aCG ◦j∗◦i Compactly Generated Stacks 19 such that

∗ a◦j ◦ℓ aCG CGHop Sh (CH) o / ShCG (CGH) o / Set aCG ◦j∗◦i ℓ is a factorization of ∗ a◦j CGHop Sh (CH) o / Set j∗◦i (up to natural isomorphism).

Note that the essential image of ℓ is the same as the essential image of j∗ ◦ i. CGHop CG ∗ Hence, a presheaf F in Set is a -sheaf if and only if the unit η of a◦j ⊥ j∗ ◦i is an isomorphism at F . We have the following immediate corollary. Corollary 3 .2. The Grothendieck topology CG is subcanonical. Lemma 3 .3. If F is a CG -sheaf, then j∗ (F ) is a K -sheaf.

Proof. Since F is a CG -sheaf, it is in the essential image of j∗ ◦ i, hence, via ηF , we have ∗ F =∼ j∗aj (F ) . By applying j∗ we have ∗ ∗ ∗ j (F ) =∼ j j∗aj (F ) . ∗ Since the co-unit ε (j) of j ⊥ j∗ is an isomorphism, this yields j∗ (F ) =∼ aj∗ (F ) .

CGHop Corollary 3 .3. If F is a presheaf in Set ,

∗ aCG (F ) =∼ j∗a (j F ) . If F ∈ Sh (CGH) is a sheaf in the open cover topology then its CG -sheaﬁﬁcation ∗ is given by j∗j F . Proof. ∗ ∗ j∗a (j F ) =∼ (j∗ ◦ i) ◦ a (j F ) , ∗ so j∗a (j F ) is in the image of j∗ ◦i, hence a CG -sheaf. Now, let G be any CG -sheaf. Then:

op ∗ ∼ op ∗ ∗ HomSetCGH (j∗a (j F ) , G) = HomSetCH (aj (F ) , j (G))

∼ op ∗ ∗ = HomSetCH (j (F ) , j (G))

∼ op ∗ = HomSetCGH (F, j∗j (G))

∼ op = HomSetCGH (F, G) .

We end this subsection by noting ordinary sheaves and CG -sheaves agree on locally compact Hausdorﬀ spaces: ∗ Let ς denote the unit of the adjunction j ⊥ j∗. 20 David Carchedi

Proposition 3 .4. Let F ∈ Sh (CGH) be a sheaf in the open cover topology and X in LCH a locally compact Hausdorﬀ space. Then the map ∗ ςF (X): F (X) → j∗j F (X) =∼ aCG (F ) (X) is a bijection. In particular, F and aCG (F ) agree on locally compact Hausdorﬀ spaces. Proof. This follows immediately from Proposition 3 .3 3 .2. Stacks for the Compactly Generated Grothendieck Topology. Denote byyCH ˆ the 2-functor CHop yCHˆ : CGH → Gpd induced by the inclusion CHop CHop . id) : Set ֒→ Gpd)(·) ˆ∗ ˆ Then, it produces a 2-adjoint pair, which we will denote by j ⊥ j∗, by constructing jˆ∗ as the weak left Kan extension ofyCH ˆ , and by letting

ˆ Y op Y j∗ (X) := HomGpdCH (yCH ˆ (X) , ) . By setting jˆ∗ (X ) (T ) := X (T ), we get a 2-functor which is weak colimit preserving and whose restriction to representables is the same asyCH ˆ , hence, by uniqueness, the above equation for jˆ∗ must be correct. Note that the co-unit is an equivalence, hence jˆ∗ is fully faithful. Similarly, denote again byyCGH ˆ the 2-functor CGHop yCGHˆ : CH → Gpd induced by the inclusion CGHop CGHop . id) : Set ֒→ Gpd)(·)

The weak left Kan extension of theyCGH ˆ along the Yoneda embedding, just as before, is left 2-adjoint to j∗ and 2-categorically full and faithful. CGHop Note that for F ∈ Set we have jˆ∗ (F ) ≃ (j∗ (F ))id , CHop and similarly for G ∈ Set , id jˆ∗ (G) ≃ (j∗ (G)) . Hence, to ease notation, we shall from now on denote jˆ∗ simply by j∗ and similarly for jˆ∗. Let St (CH) denote the bicategory of stacks on CH with respect to the Grothendieck K topology . Then we have a 2-adjoint pair a ⊥ i

a CHop St (CH) o / Gpd , i where a is the stackiﬁcation 2-functor (Deﬁnition A .3) and i is the inclusion. Then, ∗ by composition, we get a 2-adjoint pair a ◦ j ⊥ j∗ ◦ i

a◦j∗ CGHop St (CH) o / Gpd . j∗◦i Compactly Generated Stacks 21

Deﬁnition 3 .2. A stack with respect to the Grothendieck topology CG on CGH will be called a CG -stack. CGHop Let StCG (CGH) denote the full sub-bicategory of Gpd consisting of CG - stacks, and let aCG denote the associated stackiﬁcation 2-functor, and ℓ the inclu-

sion, so aCG ⊥ ℓ. Just as before, since every open covering is a CG -cover,

StCG (CGH) ⊂ St (CGH) . The following results and their proofs follow naturally from those of the previous section when combined with the Comparison Lemma for stacks, a straightforward stacky analogue of the theorem in [1] III: Corollary 3 .4. There is an equivalence of bicategories

a◦j∗◦ℓ St (CH) o / StCG (CGH) , aCG ◦j∗◦i such that

∗ a◦j ◦ℓ aCG CGHop St (CH) o / StCG (CGH) o / Gpd aCG ◦j∗◦i ℓ is a factorization of a◦j∗ CGHop St (CH) o / Gpd j∗◦i (up to natural equivalence). Lemma 3 .4. If X is a CG -stack, then j∗ (X ) is a K -stack.

op Corollary 3 .5. If X is a weak presheaf in GpdCGH ,

∗ aCG (X ) ≃ j∗a (j X ) . If X ∈ St (CGH) is a stack in the open cover topology then its CG -stackiﬁcation is ∗ given by j∗j X .

∗ Again, let ς denote the unit of the 2-adjunction j ⊥ j∗. Proposition 3 .5. Let X ∈ St (CGH) be a stack in the open cover topology and X in LCH a locally compact Hausdorﬀ space. Then the map ∗ ςX (X): X (X) → j∗j X (X) ≃ aCG (X ) (X) is an equivalence of groupoids. In particular, X and aCG (X ) agree on locally compact Hausdorﬀ spaces.

4 . Compactly Generated Stacks 4 .1. Compactly Generated Stacks. Deﬁnition 4 .1. A compactly generated stack is a presentable CG -stack (see Deﬁnition B .17). 22 David Carchedi

We denote the full sub-bicategory of StCG (CGH) of compactly generated stacks by CG TSt. Intrinsically, a compactly generated stack is a CG -stack X such that there exists a compactly generated Hausdorﬀ space X and a representable CG -epimorphism p : X → X . The map above is a CG -atlas for X . Let CHop y˜CH : CGHGpd → Gpd denote the 2-functor id G 7→ HomCGHGpd (·) , G .

Given a topological groupoid G in CGHGpd, denote by [G]K the associated K -stack a ◦ y˜CH (G). ′ CHop Let CG TSt denote the essential image in Gpd of a ◦ y˜CH, i.e., it is the full sub-bicategory of consisting of K -stacks equivalent to [G]K for some compactly generated topological groupoid G. It is immediate from Theorem 3 .4 that this bicategory is equivalent to CG TSt. In fact, the functor j∗ restricts to an equivalence ′ j∗|CG TSt′ : CG TSt → CG TSt of bicategories. Hence we have proven: Theorem 4 .1. The bicategory of compactly generated stacks, CG TSt, is equivalent to the essential image of ∗ j |TSt : TSt → St (CH) . Note that from Theorems B .1 and B .2, CG TSt is also equivalent to the bi- −1 category of fractions CGHGpd WCG of compactly generated Hausdorﬀ topological groupoids with inverted CG -Morita equivalences, and also to the bicategory CG Bun CGHGpd of compactly generated Hausdorﬀ topological groupoids with left CG -principal bundles as morphisms: Theorem 4 .2. The 2-functor

aCG ◦ y˜ : CGHGpd → CG TSt induces an equivalence of bicategories −1 ∼ PCG : CGHGpd WCG −→ CG TSt. Theorem 4 .3. The 2-functor

aCG ◦ y˜ : CGHGpd → CG TSt induces an equivalence of bicategories ′ CG ∼ PCG : Bun CGHGpd −→ CG TSt.

We note that the principal bundles in BunCG CGHGpd have a very simple description: Recall that our notion of principal bundle depends on a Grothendieck topology. When the projection map of a principal bundle admits local sections (with respect to the open cover topology), it is called ordinary. Compactly Generated Stacks 23

Proposition 4 .1. If G is a topological groupoid in CGHGpd, X is a compactly generated Hausdorﬀ space, and

G1 P ⑧⑧ µ ⑧⑧ ⑧⑧ π ⑧⑧ G0 X is a left G-space over π, then it is a CG -principal bundle if and only if the restriction of P to K is an ordinary principal G-bundle over K, for every compact subset K ⊆ X. Proof. Suppose that we are given a left G-space π : P → X, whose restriction to K is an ordinary principal G-bundle over K, for every compact subset K ⊆ X. Then, i Nα for each compact subset Kα ⊆ X, we can choose an open cover Uα ֒→ Kα i=1 over which P admits local sections. Then P admits local sections with respect to the CG -cover i . U := Uα ֒→ X The converse is trivial.

Corollary 4 .1. If G and H are topological groupoids in CGHGpd and H0 is locally compact Hausdorﬀ, then

CG BunG (H) ≃ BunG (H) , CG CG where BunG (H) denotes the groupoid of -principal G-bundles over H, and BunG (H) denotes the groupoid of ordinary principal G-bundles over H. Equivalently: If X and Y are topological stacks, and Y admits a locally compact Hausdorﬀ atlas Y → Y , then the map

HomSt(CGH) (Y , X ) → HomSt(CGH) (aCG (Y ) ,aCG (X )) X X induced by the unit ςX : → aCG ( ), and the 2-adjunction aCG ⊥ ℓ is an equivalence of groupoids. Corollary 4 .2. The CG -stackification functor restricted to the sub-bicategory of topological stacks consisting of those topological stacks which admit a locally compact Hausdorff atlas, is 2-categorically full and faithful. Theorem 4 .4. The bicategory of compactly generated stacks is equivalent to the sub-bicategory of topological stacks consisting of those topological stacks which admit a locally compact Hausdorff atlas.5 Proof. Denote the sub-bicategory of topological stacks consisting of those topological stacks which admit a locally compact Hausdorff atlas by TStLCH. Note that the image of

aCG |TStLCH : TStLCH → StCG (CGH) lies entirely in CG TSt. By Corollary 4 .2 this 2-functor is full and faithful. It suﬃces to show it is essentially surjective. Notice that the essential image is those compactly generated stacks which admit a locally compact Hausdorﬀ atlas. To

5When we work over compactly generated Hausdorﬀ spaces 24 David Carchedi complete the proof, note that if X → X is any atlas of a compactly generated stack, then the inclusion of all compact subsets of X is a CG -cover, hence

Kα → X → X α a is a CG -atlas for X which is locally compact Hausdorﬀ. Theorem 4 .5. The bicategory of compactly generated stacks has arbitrary products.

Proof. Let Xi be an arbitrary family of compactly generated stacks. Then we can choose topological groupoids Gi in CGHGpd such that

Xi ≃ [Gi]CG . Note that

[Gi]CG ≃ j∗ [Gi]K .

In light of Lemma 2 .3, we may assume without loss of generality that each Gi is ﬁbrant. Under this assumption, by Theorem 2 .4, it follows that

[Gi]CG ≃ j∗y˜CH (Gi) .

Note that the product Xi is a CG -stack, as any bicategory of stacks is complete. i It suﬃces to show thatQ this product is still presentable. Recall thaty ˜CH preserves small weak limits. Moreover, j∗ does as well as it is a right 2-adjoint. Hence

Xi ≃ j∗y˜CH (Gi) ≃ j∗y˜CH Gi . i i i ! Y Y Y It follows that

Xi ≃ aCG Xi ≃ aCG j∗y˜CH Gi i i ! i !! Y Y Y ∗ ∗ ≃ (j∗ ◦ a ◦ j ) ◦ j∗ ◦ (j y˜) Gi i ! Y ∗ ≃ (j∗ ◦ a ◦ j ) ◦ (˜y) Gi i ! Y

≃ Gi . " i #CG Y

Corollary 4 .3. The bicategory of compactly generated stacks is closed under arbitrary small weak limits. Proof. Since CGHGpd is closed under binary weak fibered products and the stackification 2-functor aCG preserves finite weak limits, the bicategory CG TSt is closed under binary weak fibered products. By Theorem 4 .5, this bicategory has arbitrary Compactly Generated Stacks 25 small products. Since CG TSt is a (2, 1)-category, by [10] it follows that CG TSt has all limits and hence is complete.

4 .2. Mapping Stacks of Compactly Generated Stacks. Recall that if X and Y are any stacks over CGH, they have a mapping stack

M Y X op Y X ap ( , ) (T ) = HomGpdCGH ( × T, ) . It is the goal of this section to prove that if X and Y are compactly generated stacks, then so is Map (Y , X ).

Lemma 4 .6. If Y ≃ [H]CG is a compactly generated stack with H0 compact Haus- dorﬀ, and X ≃ [G]CG an arbitrary compactly generated stack, then Map (Y , X ) is a compactly generated stack. More speciﬁcally, if K is a presentation for the topological stack Map ([H] , [G]) ensured by Theorem 2 .8, then

Map (Y , X ) ≃ [K]CG . In particular, Map (Y , X ) and Map ([H] , [G]) restrict to the same stack over locally compact Hausdorff spaces. Proof. Since any CG -stack is completely determined by its restriction to CH, it suffices to show that for any compact Hausdorff space T ∈ CH,

op Y X [K]CG (T ) ≃ HomGpdCGH ( × T, ) . But, since T is compact Hausdorﬀ

[K]CG (T ) ≃ [K] (T ) and because of the deﬁnition of K

[K] (T ) ≃ HomGpdCGHop ([H] × T, [G]) .

Furthermore, since Y × T ≃ [H× T ]CG and H× T has compact Hausdorﬀ object space, by Corollary 4 .1,

HomGpdCGHop ([H] × T, [G]) ≃ HomGpdCGHop ([H]CG × T, [G]CG )

≃ HomGpdCGHop (Y × T, X ) .

Hence

[K]CG (T ) ≃ Hom (Y × T, X ) .

Corollary 4 .4. If K is a compact Hausdorﬀ space and X an arbitrary compactly generated stack, then Map (K, X ) is a compactly generated stack. Lemma 4 .7. If X is a compactly generated Hausdorﬀ space and X an arbitrary compactly generated stack, then Map (X, X ) is a compactly generated stack.

iα Proof. Let Kα ֒→ X denote the inclusion of all compact subsets of X. This is a CG -cover for X. Let Y be an arbitrary compactly generated Hausdorﬀ space. By Proposition A .1, we have that in CG TSt 26 David Carchedi

X ≃ holim Kα −−−−−−−→ Kα֒→X and

X × Y ≃ holim (Kα × Y ) , −−−−−−−→ Kα×Y֒→X×Y the latter since colimits are universal in any bicategory of stacks. Hence

Map (X, X ) (Y ) ≃ HomGpdCGHop (X × Y, X )

op X ≃ HomGpdCGH holim (Kα × Y ) ,  −−−−−−−→  Kα×Y֒→X×Y

 op  ≃ holim HomGpdCGH (Kα × Y, X ) ←−−−−−−− Kα×Y֒→X×Y

op M X ≃ holim HomGpdCGH (Y, ap (Kα, )) ←−−−−−−− Kα×Y֒→X×Y

≃ HomGpdCGHop Y, holim Map (Kα, X )  ←−−−−−−−  Kα֒→X  

≃ holim Map (Kα, X ) (Y ) . ←−−−−−−−  Kα֒→X   Therefore

Map (X, X ) ≃ holim Map (Kα, X ) . ←−−−−−−− Kα֒→X So by Corollary 4 .3, Map (X, X ) is a compactly generated stack.

Theorem 4 .8. If X and Y are arbitrary compactly generated stacks, then Map (Y , X ) is a compactly generated stack. Proof. Let Y be presented by a topological groupoid H. By Lemma B .3, we can write Y as the weak colimit of the following diagram: H × H // H / H , 1 H0 1 / 1 / 0 where the three parallel arrows are the ﬁrst and second projections and the composition map. Furthermore, let X be any compactly generated Hausdorﬀ space. Then Y × X is the weak colimit of

(H × H ) × X // H × X / H × X. 1 H0 1 / 1 / 0 With this in mind, in much the same way as Lemma 4 .7, some simple calcu- lations allow one to express Map (Y , X ) as a weak limit of a diagram involving M X M X M X ap (H1 ×H0 H1, ), ap (H1, ), and ap (H0, ) , all of which are compactly generated stacks by Lemma 4 .7. Compactly Generated Stacks 27

We presented the proof of Cartesian-closure in this way to emphasize the role of completeness and compact generation. We will now give a concrete description of the mapping stack of two compactly generated stacks. Note that since the inclusion of all compact subsets of a space is a CG -cover, every compactly generated stack has a locally compact, paracompact Hausdorﬀ atlas.

Theorem 4 .9. Let X ≃ [G]CG and Y ≃ [H]CG be two compactly generated stacks. Assume (without loss of generality) that H0 is locally compact and paracompact Hausdorff. Then Fib (G)H ≃ Map (Y , X ) . CG h i Proof. It suffices to check that Fib (G)H and Map (Y , X ) agree on every com- CG pact Hausdorff space T . Followingh the samei proof as Theorem 2 .9, one only has to realize that the product of a compact Hausdorff space with a paracompact Haus- dorff space is paracompact Hausdorff [16]. The rest of the proof is identical.

Remark. This implies that the bicategory of topological stacks (in compactly generated Hausdorff spaces) with locally compact Hausdorff atlases is Cartesian closed. This might seem surprising since, after all, locally compact Hausdorff spaces are quite far from being Cartesian closed. What is happening is that the exponential of two locally compact Hausdorff spaces is not a space in this description, but a stack! (This stack is actually a sheaf.) In fact, the category of compactly generated Hausdorff spaces embeds into this bicategory by sending a space X to the stack as- (id) CG sociated to the topological groupoid (X)K , where K denotes the -cover which is the inclusion of all compact subsets of X. 4 .3. Homotopy Types of Compactly Generated Stacks. In [20], Noohi con- structs a functorial assignment to each topological stack X , a weak homotopy type. For X a topological stack, its weak homotopy type turns out to be the weak homotopy type of kGk for any topological groupoid G for which X ≃ [G]. Moreover, each topological stack X admits at atlas which is also a weak homotopy equivalence; the canonical atlas ϕ : kGk → X coming from Lemma 2 .3 is a weak homotopy equivalence. A particular corollary is: Corollary 4 .5. If G → H is a Morita equivalence, the induced map kGk → kHk is a weak homotopy equivalence. This is a classical result. For instance, it is proven for the case of étale topological groupoids in [13] and [14]. 6 In this section, we extend these results to the setting of compactly generated stacks. We begin with a technical notion of a shrinkable map, which will prove quite useful.

6I would like to thank Ieke Moerdijk for explaining to me how to extend his method of proof to any topological groupoid whose object and arrow spaces have a basis of contractible open sets. 28 David Carchedi

Deﬁnition 4 .2. [5] A continuous map f : X → B is shrinkable if admits a section s : B → X together with a homotopy H : I × X → X from sf to idX over B, i.e. for all t, the map

Ht : X → X is a map in Top/B from f to f. Remark. Every shrinkable map is in particular a homotopy equivalence. Definition 4 .3. A continuous map f : X → B is locally shrinkable [20] if there exists an open cover (Uα) of B such that for each α, the induced map −1 f|Uα : f (Uα) → Uα is shrinkable. A map f : X → B is called CG -locally shrinkable if there exists a CG -cover (Vi) of B such that the same condition holds. Clearly, shrinkable ⇒ locally shrinkable ⇒ CG -locally shrinkable. Definition 4 .4. A continuous map f : X → B is quasi-shrinkable if for every map T → B from a locally compact, paracompact Hausdorff space T, the induced map X ×Y T → T is shrinkable. Lemma 4 .10. Every CG -locally shrinkable map is quasi-shrinkable. Proof. Since every CG -cover of a locally compact Hausdorff space can be refined by an open one, and every open cover of a paracompact Hausdorff space can be refined by a numerable one, the result follows from [5], Corollary 3.2. Definition 4 .5. A map f : X → Y of spaces is a universal weak equivalence if for any map T → Y , the induced map T ×Y X → T is a weak homotopy equivalence. The following lemma is a useful characterization of universal weak equivalences: Lemma 4 .11. A map f : X → B is a universal weak equivalence if and only if, for n n n all n ≥ 0, for any map D → B from the n-disk, the induced map X ×B D → D n is a weak homotopy equivalence (i.e. X ×B D is weakly contractible). Proof. One direction is clear by definition. Conversely, let f : X → B be a map satisfying the stated hypothesis for each n-disk. We will first show that f is a weak equivalence. Denote by I the unit interval [0, 1] , and let E (f) denote the homotopy fiber of f, E (f) / BI

ev(0) f X / B, where ev(0) is evaluation at 0. We may factor X → B as ev(1) X → E (f) −−−−−−→ B Compactly Generated Stacks 29 where X → E (f) is a homotopy equivalence and the evaluation map (at 1) is a ﬁbration. From the long exact sequence of homotopy groups resulting from the ﬁber sequence

E (f)b → Ef → B, it suﬃces to show that for each b ∈ B in the image of f, the homotopy ﬁber −1 E (f)b = ev(1) (b) is weakly contractible. n−1 n Suppose we are given a based map l : S → E (f)b . Identifying D with the cone on Sn−1, this is the same as giving a commutative diagram

Sn−1 / Dn

l1 l2 f X / B, where Sn−1 → Dn is the canonical map. Let f˜ denote the induced map

˜ n f n X ×B D / D

l2 f X / B. By our hypothesis, f˜ is a weak homotopy equivalence. Moreover, it is easy to check that the map n−1 l : S → E (f)b factors through the canonical map

q : E(f˜)b → E(f)b, say l = ql′ for some ′ n−1 l : S → E(f˜)b. ′ As f˜ is a weak equivalence, E(f˜)b is weakly contractible. So, l is null-homotopic, and hence so is l. It follows that E (f)b is also weakly contractible, and thus f is a weak equivalence. Moreover, f is in fact a universal weak equivalence since if T → X is any map, the induced map X ×B T satisﬁes the same hypothesis that f does. Corollary 4 .6. Every quasi-shrinkable map is a universal weak equivalence. Deﬁnition 4 .6. A representable map X → Y of stacks on CGH (with respect to either the CG -topology or the open cover topology) is said to be shrinkable, locally- shrinkable, CG -locally shrinkable, quasi-shrinkable, or a universal weak equivalence, if and only if for every map T → X from a topological space, the induced map X ×Y T → T is. ′ Lemma 4 .12. Let f : X → Y and g : Y → Y be morphisms in StCG (CGH) such that g is a CG -epimorphism and the induced map ′ ′ X ×Y Y → Y is a representable CG -locally shrinkable map. Then f is also representable and CG -locally shrinkable. 30 David Carchedi

Proof. Let h : T → Y be arbitrary. Choose a CG -cover (Vα) of T such that for all α, there is a 2-commutative diagram

Vα / T

hα h g Y ′ / Y . Note, by assumption, the induced maps

X ×Y Vα → Vα are CG -locally shrinkable maps of topological spaces. By reﬁning this CG -cover if necessary, we can arrange that each of these maps is in fact shrinkable. It follows that X ×Y T is a topological space, and that the collection (X ×Y Vα) is a CG - cover of it. Since the restriction of

X ×Y T → T to each element of this cover is

X ×Y Vα → Vα, it follows that X ×Y T → T is CG -locally shrinkable.

Theorem 4 .13. Let X ≃ [G]CG be a compactly generated stack. Then the atlas kGk → X is CG -locally shrinkable. Proof. In [20], it is shown that we have a 2-Cartesian diagram

f kEGk / G0

ϕ kGk / [G] with f shrinkable. Now, the stackiﬁcation 2-functor aCG commutes with ﬁnite weak limits, hence, the following is also a 2-Cartesian diagram:

f kEGk / G0

ϕ¯ kGk / X .

Since the map G0 → X is a CG -epimorphism and f is shrinkable, it follows from Lemma 4 .12 thatϕ ¯ is CG -locally shrinkable.

Corollary 4 .7. Let X ≃ [G]CG be a compactly generated stack. Then the atlas kGk → X is a universal weak equivalence. Proof. This follows immediately from Lemma 4 .10 and Corollary 4 .6. Compactly Generated Stacks 31

Corollary 4 .8. Let φ : G → H be a CG -Morita equivalence between two topological groupoids G and H. Then φ induces a weak homotopy equivalence

kφk : kGk → kHk .

Proof. Let X := [G]CG ≃ [H]CG . Then each atlas kGk → X and kHk → X is a universal weak equivalence. The following diagram 2-commutes (with the outer square Cartesian):

β kGk ×X kHk / kHk 8 α kϕk kGk / X . Since each atlas is a universal weak equivalence, α and β are weak equivalences, and hence so is kϕk.

Example 1. Let X be a topological space. Consider the inclusion of all its compact subsets (Kα ֒→ X). This is a CG -cover, so the associated groupoid

K = Kα ∩ Kβ ⇒ Kα is CG -Morita equivalent to X.a It follows froma Corollary 4 .8 that kKk is weakly homotopy equivalent to X. We now copy Noohi in [20] to give a functorial assignment to each compactly generated stack a weak homotopy type. Given a bicategory C , denote the 1-category obtained by identifying equivalent 1-morphisms by τ1 (C ) . Suppose we are given a full sub-bicategory B of C which is in fact (equivalent to) a 1-category, and is closed under pullbacks. For example, consider C to be the bicategory of compactly generated stacks, CG TSt, and for B to the category of compactly generated Hausdorﬀ spaces CGH. Let R be a class of morphisms in B which contains all isomorphisms, and is stable under pullback. Let R˜ denote the class of morphisms f : X → Y in C such that for every h : T → Y , with T in B, the weak pullback X ×Y T is in B and the induced map

X ×Y T → T is in R. In the case where C = CG TSt and B = CGH, then f is a representable map with property R. Lemma 4 .14. [20] In the set up just described, if for every object X of C there exists a morphism ϕ (X ):Θ(X ) → X in R˜ from an object Θ (X ) of B, then there is an induced adjunction

y ˜−1 C o −1 R τ1 ( ) o / R B Θ

with y ⊥ Θ, and with y fully faithful. Moreover, the components of the co-unit of this adjunction are in R.˜ 32 David Carchedi

Theorem 4 .15. There exists a functor Ω: CG TSt → Ho (TOP) assigning to each compactly generated stack X , a weak homotopy type. Moreover, for each X , there is a CG -atlas X → X , which is a universal weak equivalence from a space X whose homotopy type is Ω (X ) . Proof. In the previous lemma, let C = CG TSt, B = CGH, and let R be the class of universal weak equivalences. Use Corollary 4 .7 to pick for each compactly generated stack X , a CG -atlas which is a universal weak equivalence. Lemma 4 .14 implies that there is an induced adjunction y ˜−1 CG o −1CGH R τ1 ( TSt) o / R , Θ where the unit of this adjunction is an equivalence, and the co-unit for X , yΘ (X ) → X , is the chosen atlas. Let S denote the class of weak homotopy equivalences in CGH. Let T denote y (S) . Then, since y (S)= T, and Θ (T )= S, it follows that there is an induced adjunction

y¯ −1 ˜−1 CG o −1 −1CGH S R τ1 ( TSt) o / S R . Θ¯ Note that there are canonical equivalences

−1 −1 −1 S R˜ τ1 (CG TSt) ≃ S τ1 (CG TSt) , and S−1 R−1CGH ≃ S−1CGH. Moreover, since every topological space has the weak homotopy type of a compactly generated Hausdorﬀ space, S−1CGH is equivalent to the homotopy category of spaces, Ho(TOP) . Note that the following diagram 2-commutes

−1 Θ −1 R˜ τ1 (CG TSt) / R CGH ♦7 ♦♦♦ ♦♦♦ ♦♦♦ ♦♦ ¯ −1 Θ CG TSt / S τ1 (CG TSt) / Ho (TOP) . Denote either (naturally isomorphic) composite by Ω : CG TSt → Ho (TOP) . 4 .4. Comparison with Topological Stacks. In this subsection, we will extend the results of the previous section to give a functorial assignment of a weak homotopy type to a wider class of stacks, which include all compactly generated stacks and all topological stacks. We will then show that for a given topological stack X , the induced map to its associated compactly generated stack aCG (X ) is a weak homotopy equivalence. Finally, we will show in what sense compactly generated stacks are to topological stacks what compactly generated spaces are to topological spaces (Theorem 4 .20). Compactly Generated Stacks 33

Proposition 4 .2. For a stack X over compactly generated Hausdorﬀ spaces (with respect to the open cover topology) whose diagonal ∆ : X → X × X is representable, the following conditions are equivalent: i) the CG -stackiﬁcation of X is a compactly generated stack, ii) there exists a topological space X and a morphism X → X such that, for all spaces T , the induced map

T ×X X → T admits local sections with respect to the topology CG (i.e. is a CG -covering morphism; See Deﬁnition B .19), iii) there exists a topological space X and a morphism X → X such that, for all compact Hausdorﬀ spaces T , the induced map

T ×X X → T admits local sections (i.e. is an epimorphism), iv) there exists a topological space X and a morphism X → X such that, for all locally compact Hausdorﬀ spaces T , the induced map

T ×X X → T admits local sections (i.e. is an epimorphism), v) there exists a topological stack X¯ and a map q : X¯ → X such that, for all locally compact Hausdorff spaces T, q (T ): X¯ (T ) → X (T ) is an equivalence of groupoids. Proof. Suppose that condition i) is satisfied. Note that condition ii) is equivalent to saying that there exists a CG -covering morphism X → X from a topological space (See Definition B .19.) Let

p : X → aCG (X ) be a locally compact Hausdorff atlas for the compactly generated stack aCG (X ) . Then it factors (up to equivalence) as x X −→ X → aCG (X ) , for some map x : X → X . Note that the CG -stackification of x is (equivalent to) p, hence is an epimorphism in StCG (CGH) . This implies x is a CG -covering morphism. So i) ⇒ ii). Since any CG -cover of a locally compact Hausdorff space can be refined by an open one, ii) (⇒ iv)) ⇒ iii). From Corollary 3 .4, it follows that there is an equivalence of bicategories

Λ : St (CH) → StCG (CGH) , such that for every stack X on CGH with respect to the open cover topology, ∗ Λ (j X ) ≃ aCG (X ) . Hence iii) (⇒ ii)) ⇒ iv). 34 David Carchedi

Suppose iv) holds. Then iv) ⇒ iii) trivially, and iii) ⇒ ii) by the above ar- gument. So there exists a CG -covering map X → X . Hence, the induced map X → aCG (X ) is an epimorphism (in particular, this implies i)). Consider the induced map α :[X ×X X ⇒ X] → X . It is a monomorphism, and since X → X is a CG -covering morphism, α is too. Since stackiﬁcation is left-exact, it preserves monomorphisms and hence aCG (α) is an equivalence between the compactly generated stack

[X ×X X ⇒ X]CG and aCG (X ) . Proposition 3 .5 implies that α satisﬁes v). Hence, iv) ⇒ v). Suppose that v) holds for a morphism q : X¯ → X from a topological stack. Then j∗ X¯ → j∗X is an equivalence. Hence, aCG (q) is an equivalence between aCG (X ) and the compactly generated stack aCG X¯ . Hence v) ⇒ i).

Definition 4 .7. A stack X over compactly generated Hausdorff spaces (with respect to the open cover topology) whose diagonal ∆ : X → X × X is representable, is quasi-topological if any of the equivalent conditions of Propo- sition 4 .2 hold. Denote the full sub-bicategory of St (CGH) consisting of the quasi- topological stacks by QuasiTSt. The following proposition is immediate. Proposition 4 .3. If X is a stack over compactly generated Hausdorff spaces which is topological, paratopological, or compactly generated, then it is quasi-topological. Lemma 4 .16. Let X be a quasi-topological stack, and let

h : T → aCG (X ) X be a map from a locally compact Hausdorﬀ space. Then T ×aCG (X ) is a topological space and the induced map X T ×aCG (X ) → T is a homeomorphism.

Proof. Let X → aCG (X ) be a locally compact Hausdorﬀ CG -atlas. Then it factors (up to equivalence) as x ςX X −→ X −→ aCG (X ) . Moreover, as T is locally compact Hausdorﬀ, there is a 2-commutative lift h′ X : ′ h ςX T / aCG (X ) . h Compactly Generated Stacks 35

Consider the weak pullback P / X

h′ T / X . As a sheaf, P assigns each space Z, the set of triples (f,g,α) with

f : Z → T,

g : Z → X, and α : xf ⇒ h′g. Since the diagonal of X is representable, P is represented by a compactly generated Hausdorﬀ space. Consider now the weak pullback diagram

P ′ / X

ςX ◦x

′ ςX ◦h T / aCG (X ) .

The sheaf P ′ is again representable, and it assigns each space Z the set of triples (f,g,β) with f : Z → T,

g : Z → X, and ′ β : ςX ◦ xf ⇒ ςX ◦ h g.

Consider the induced map P → P ′ given by composition with ςX . Since for every locally compact Hausdorﬀ space S, ςX (S) is an equivalence of groupoids, it follows ′ that the induced map yCH (P ) → yCH (P ) is an isomorphism, where

CHop yCH : CGH → Set is the functor which assigns a compactly generated Hausdorﬀ space X the presheaf S 7→ HomCGH (S,X) . Since yCH is fully-faithful, it follows that the induced map P → P ′ is an isomorphism. Finally, regard the following 2-commutative diagram:

P ′ / X

h′ T / X

ςX idT h T / aCG (X ) . 36 David Carchedi

The outer square is Cartesian, and so is the upper-square. It follows that

h′ T / X

ςX idT h T / aCG (X ) is Cartesian as well. Corollary 4 .9. For every quasi-topological stack X , there exists a representable universal weak equivalence Θ (X ) → X , from a topological space Θ (X ) .

Proof. Let X be a quasi-topological stack, and let X → aCG (X ) be a locally compact Hausdorﬀ CG -atlas. Then it factors (up to equivalence) as

x ςX X −→ X −→ aCG (X ) . Denote by G the topological groupoid

X ×X X ⇒ X.

There is a canonical map [G] → X and the unit map ς[G] factors as

ςX [G] → X −→ aCG (X ) . The composite ςX kGk → [G] → X −→ aCG (X ) is a representable quasi-shrinkable morphism, by Theorem 4 .13. From Lemma 4 .16, it follows that kGk → [G] → X is a representable quasi-shrinkable morphism as well, and in particular, a representable universal weak equivalence, by Corollary 4 .6. Theorem 4 .17. There exists a functor Ω: QuasiTSt → Ho (TOP) assigning to each quasi-topological stack X , a weak homotopy type. Moreover, for each X , there is a representable universal weak equivalence X → X , from a space X whose homotopy type is Ω (X ) . Furthermore, we can arrange for the functor Ω to restrict to the one of Theorem 4 .15 on compactly generated stacks. Proof. Using the notation of Lemma 4 .14, let C = QuasiTSt, B = CGH, and let R be the class of universal weak equivalences. Using the notation of the proof of Corollary 4 .9, for each quasi-topological stack X , let ϕ (X ):Θ(X )= kGk → [G] → X . The rest is identical to the proof of Theorem 4 .15. Remark. This agrees with the functorial construction of a weak homotopy type of a topological or paratopological stack given in [20], by construction. Deﬁnition 4 .8. A morphism f : X → Y in QuasiTSt is a weak homotopy equivalence if Ω (f) is an isomorphism. Compactly Generated Stacks 37

Theorem 4 .18. Let X be a quasi-topological stack. Then the unit map

ςX : X → aCG (X ) induces an equivalence of groupoids

X (Y ) → aCG X (Y ) for all locally compact Hausdorff spaces Y , and ςX is a weak homotopy equivalence. Proof. The first statement follows immediately from Proposition 3 .5. For the second, letting R denote the class of universal weak equivalences, we can factor Ω as QuasiTSt → R−1QuasiTSt −→Θ R−1CGH → Ho (TOP) . To show Ω (ςX ) is an isomorphism, it suffices to show Θ (ςX ) is. From [20], an arrow between two spaces X and Y in R−1CGH is a span (r, g) of the form T ⑧ ❄❄ r ⑧⑧ ❄❄ g ⑧⑧ ❄❄ ⑧ ❄❄ ⑧⑧ XY, with r a universal weak equivalence. Moreover, if f : X → Y is a morphism of quasi-topological stacks, Θ (f) is given by the span (w,f ′) provided by the diagram

f ′ Θ (X ) ×Y Θ (Y ) / Θ (Y )

w ϕ(Y ) ϕ(X ) f Θ (X ) / X / Y . Such a span is an isomorphism if and only if f ′ is a universal weak equivalence. It follows that Θ (ςX ) is given by the span deﬁned by the diagram

f ′ Θ (X ) ×Y Θ (aCG (X )) / Θ (aCG (X ))

w ϕ(aCG (X ))

ϕ(X ) ςX Θ (X ) / X / aCG (X ) . Notice that ϕ(X ) ςX Θ (X ) −−−−−−→ X −−−−−−→ aCG (X ) is a representable universal weak equivalence. It follows that f ′ is as well, so we are done. Corollary 4 .10. Let X be a topological or paratopological stack. Then the unit map ςX : X → aCG (X ) induces an equivalence of groupoids

X (Y ) → aCG X (Y ) for all locally compact Hausdorﬀ spaces Y . Moreover, ςX is a weak homotopy equivalence. 38 David Carchedi

In particular, to any topological stack, there is a canonically associated compactly generated stack of the same weak homotopy type which restricts to the same stack over locally compact Hausdorﬀ spaces. Conversely, if

Y ≃ [H]CG is a compactly generated stack, [H] is an associated topological stack for which the same is true. Theorem 4 .19. Let X and Y be topological stacks such that Y admits a locally compact Hausdorﬀ atlas. Then Map (Y , X ) is a paratopological stack, Map (aCG (Y ) ,aCG (X )) is a compactly generated stack, and there is a canonical weak homotopy equivalence

Map (Y , X ) → Map (aCG (Y ) ,a (X )) .

Moreover, Map (Y , X ) and Map (aCG (Y ) ,aCG (X )) restrict to the same stack over locally compact Hausdorﬀ spaces. Proof. The fact that Map (Y , X ) is a paratopological stack follows from Theorem 2 .9, and that Map (aCG (Y ) ,aCG (X )) is a compactly generated stack follows from Theorem 4 .8. To prove the rest, it suﬃces to prove that

aCG (Map (X , Y )) ≃ Map (aCG (X ) ,aCG (Y )) . For this, it is enough to show that they restrict to the same stack over compact Hausdorﬀ spaces. Let T be a compact Hausdorﬀ space. Then

aCG (Map (X , Y )) assigns T the groupoid Hom (T × X , Y ) , since it agrees with Map (X , Y ) along locally compact Hausdorﬀ spaces. From Corollary 4 .1, since T ×X admits a locally compact atlas, this is in turn equivalent to the groupoid

Hom (aCG (T × X ) ,aCG (Y )) ≃ Map (aCG (X ) ,aCG (Y )) (T ) .

We end this paper by showing exactly in what sense compactly generated stacks are to topological stacks what compactly generated spaces are to topological spaces: Recall that there is an adjunction

k CG o TOP, v / exhibiting compactly generated spaces as a co-reﬂective subcategory of the category of topological spaces, and for any space X, the co-reﬂector vk (X) → X is a weak homotopy equivalence. We now present the 2-categorical analogue of this statement: Theorem 4 .20. There is a 2-adjunction

k CG TSt o TSt, v /

v ⊥ k, Compactly Generated Stacks 39 exhibiting compactly generated stacks as a co-reflective sub-bicategory of topological stacks, and for any topological stack X , the co-reflector vk (X ) → X is a weak homotopy equivalence. A topological stack is in the essential image of the 2-functor v : CG TSt → TSt if and only if it admits a locally compact Hausdorff atlas. Proof. Let us first start with the 2-functor k : TSt → CG TSt. We define it to be the restriction of the stackification 2-functor CGHop aCG : Gpd → StCG (CGH) to TSt. Note that since every open cover is a CG -cover, for all topological groupoids G and H, there is a canonical full and faithful functor CG BunH (G) → BunH (G) . In fact, it is literally an inclusion on the level of objects. These assemble into a homomorphism of bicategories CG k˜ : BunCGHGpd → Bun CGHGpd which is the identify on objects, 1-morphisms, and 2-morphisms (but it is not 2- categorically full and faithful). Composing with the canonical equivalences,

k˜ CG TSt −→∼ BunCGHGpd −→ Bun CGHGpd −→∼ CG TSt is a factorization (up to equivalence) of aCG |TSt. We will construct a left 2-adjoint to k˜. For all topological groupoids G, letv ˜ (G) be the Cechˇ groupoid GK of G with respect to the CG -cover of G0 by all its compact subsets. In particular,v ˜ (G) has a locally compact Hausdorﬀ object space. The canonical map GK → G is a CG - k Morita equivalence. Denote the associated CG -principal G-bundle over GK by 1 G. k Since (GK)0 is locally compact Hausdorﬀ, 1 G is an ordinary principal bundle k 1G ∈ BunG (˜v (G))0 .

Since GK → G is a CG -Morita equivalence, there exists a CG -principal GK-bundle over G, rG and isomorphisms k ∼ αG : rG ⊗ 1 G −→ idG ∼ k βG : idGK −→ 1 G ⊗ rG , where the tensor product symbol ⊗ denotes composition in the bicategory of principal bundles. The assignment G 7→ v˜ (G) = GK extends to a homomorphism of bicategories CG k v˜ : Bun CGHGpd −→ BunCGHGpd, 40 David Carchedi by CG v˜H,G : BunG (H)0 → Bunv˜(G) (˜v (H)) k P 7→ rH ⊗ P ⊗ 1 G, and similarly on 2-cells. Note that for G and H topological groupoids, there is a natural equivalence of groupoids

Hom(˜v (G) , H) = BunH (GK) CG = BunH (GK) (since GK has locally compact Hausdorﬀ object space) CG CG ≃ BunH (G) (since GK is -Morita equivalent to G) = Hom (G, k (H)) , which sends a principal G-bundle P over HK to P ⊗ rG . An inverse for this equivalence is given by sending CG P ∈ BunG (H) k ˜ to P ⊗ 1H. These equivalences deﬁne an adjunction of bicategories,v ˜ ⊥ k. The unit of this adjunction is given by CG rG : G → GK = k˜v˜ (G) in Bun CGHGpd, which is an equivalence. It follows that v is bicategorically full and faithful. The co-unit is given by k 1 G :˜vk˜ (G)= GK → G in BunCGHGpd. The induced adjunction

˜ CG k CG TSt ≃ Bun CGHGpd o / GpdBunCGH ≃ TSt, v˜

is the desired v k.⊥ The 2-functor v sends a compactly generated stack equivalent to [G]CG to [GK] , which has a locally compact Hausdorff atlas. From the general theory of adjunctions, the essential image is precisely the sub-bicategory of topological stacks for over which k = aCG restricts to a full and faithful 2-functor. We have already shown that the essential image is contained in those topological stacks which admit a locally compact Hausdorff atlas. By Corollary 4 .2, aCG restricted to this sub-bicategory is full and faithful, hence the essential image of v is topological stacks which admit a locally compact Hausdorff atlas. It remains to show that the co-unit is a weak homotopy equivalence. Let [G] be a topological stack. Then the co-unit is given by the canonical map

ε[G] :[GK ] → [G] . Notice that the following diagram is 2-commutative:

ε[G] [GK] / [G] ❏❏ ❏❏ ❏ ς[G] ς ❏❏ [GK] ❏ ❏% aCG ([G]) .

By Corollary 4 .10, the maps ς[G] and ς[GK] are weak homotopy equivalences. It follows that so is ε[G]. Compactly Generated Stacks 41

A. Stacks Let (C ,J) be a small Grothendieck site and let Gpd denote the bicategory of op groupoids7. Denote by GpdC the bicategory of weak presheaves C op → Gpd. That is, weak contravariant 2-functors from C to Gpd. We call this the bicategory of weak presheaves on C . There exists a canonical inclusion C op C op (·)id : Set → Gpd , where, each presheaf F is sent to the weak presheaf which assigns to each object C the category (F (C))id whose objects are F (C) and whose arrows are all identities. id If C is an object of C , we usually denote (yC) simply by C. We also have a version of the Yoneda lemma: Lemma A .1. [6] The 2-Yoneda Lemma: If C is an object of C and X a weak presheaf, then there is a natural equivalence of groupoids

HomGpdCop (C, X ) ≃ X (C) .

op Remark. The bicategory GpdC is both complete and cocomplete; weak limits are computed “pointwise”:

holim Xi (X) = holim Xi (X) ←−−−−−−− ←−−−−−−− where the weak limit to the right is computed in Gpd. Similarly for weak colimits. Deﬁnition A .1. A weak presheaf X is called a stack if for every object C and covering sieve S, the natural map

op X op X HomGpdC (C, ) → HomGpdC (S, ) is an equivalence of groupoids. If this map is fully faithful, X is called separated (or a prestack). Although it is not standard, if this map is faithful, we will call it weakly separated.

op We denote the full sub-bicategory of GpdC consisting of those weak presheafs that are stacks by StJ (C ). It is immediate from the deﬁnition that the weak limit of any small diagram of stacks is again a stack. If B is a basis for the topology J, then it suﬃces to check this condition for every sieve of the form SU , where U is a covering family. Namely, a weak presheaf is a stack if and only if for every covering family U = (Ui → C)i the induced map → X (C) → holim X (Ui) → X (Uij ) → X (Uijk) ←−−−−−−− → → is an equivalence of groupoids.hY Y Y i If this map is fully faithful, X is is separated and if this map is faithful, it is weakly separated. The associated groupoid → Des (X , U) := holim X (Ui) → X (Uij ) → X (Uijk) , ←−−−−−−− → → hY Y Y i 7Technically, we mean those groupoids which are equivalent to a small category. 42 David Carchedi obtained as weak limit of the above diagram of groupoids, is called the category of descent data for X at U.

Proposition A .1. If J is subcanonical and (Ui → C)i is a covering family for an object C, then, in the bicategory of stacks → C ≃ holim Uijk → Uij → Ui −−−−−−−→ → → ha a a i We will often simply write C ≃ holim Ui. −−−−−−−→ Ui→C Deﬁnition A .2. A morphism ϕ : X → Y of stacks is called representable if for any object C ∈ C and any morphism C → Y , the weak pullback C ×Y X in the category of stacks is (equivalent to) an object D of C . We can now deﬁne a 2-functor op op (·)+ : GpdC → GpdC by

X + op (C) := holim HomGpdC (S, F ) . −−−−−−−→ S∈Cov(C) We can alternatively deﬁne (·)+ by the equation

X + (C) := holim Des (X , U) −−−−−−−→ U∈cov(C) and obtain a naturally equivalent 2-functor. Remark. The weak colimit in either definition must be indexed over a suitable bicategory of covers. X is separated if and only if the canonical map X → X + is a fully faithful, and weakly separated if and only if it is faithful. X is a stack if and only if this map is an equivalence. If X is separated, then X + is a stack, and if X is only weakly separated, then X + is separated. Furthermore, X + is always weakly separated, for any X . Hence, X +++ is always a stack. +++ Definition A .3. We denote by aJ the 2-functor X 7→ X . It is called the stackification 2-functor . + If X is separated, aJ (X ) ≃ X , and if X is a stack, aJ (X ) ≃ X . The 2-functor aJ is left 2-adjoint to the inclusion C op i : StJ (C ) ֒→ Gpd and preserves finite weak limits.

Remark. StJ (C ) is both complete and cocomplete. Since i is a right adjoint, it follows that the computation of weak limits of stacks can be done in the category op GpdC , hence can be done “pointwise”. To compute the weak colimit of a diagram op of stacks, one must ﬁrst compute it in GpdC and then apply the stackiﬁcation functor aJ . Compactly Generated Stacks 43

We end by remarking that the bicategory StJ (C ) is Cartesian closed. The exponent X Y of two stacks is given by

Y X (C) = HomGpdCop (Y × C, X ) , and satisﬁes Y HomGpdCop Z , X ≃ HomGpdCop (Y × Z , X ) for all stacks Z .

B. Topological Groupoids and Topological Stacks B .1. Deﬁnitions and Some Examples. This section is a review of topological groupoids and topological stacks. The material is entirely standard aside from the fact that some of the standard results are stated for more general Grothendieck topologies. Deﬁnition B .1. A topological groupoid is a groupoid object in TOP, the category of topological spaces. Explicitly, it is a diagram

i s m / G1 ×G0 G1 / G1 / G0 Y t

u of topological spaces and continuous maps satisfying the usual axioms. Forgetting the topological structure (i.e. applying the forgetful functor from TOP to Set), one obtains an ordinary (small) groupoid. Topological groupoids form a bicategory with continuous functors as 1-morphisms and continuous natural transformations as 2-morphisms. We denote the bicategory of topological groupoids by TOPGpd. Remark. TOPGpd has weak fibered products and arbitrary products. Since it is a (2, 1)-category, by [10] it follows that it is a complete bicategory. Definition B .2. Given a topological space X, we denote by (X)id the topological groupoid whose object and arrow space are both X and all of whose structure maps are the identity morphism of X. The arrow space is the collection of all the identity arrows for the objects X. Definition B .3. Given a topological space X, the pair groupoid Pair (X) is the topological groupoid whose object space is X and whose arrow space is X × X, where an element (x, y) ∈ X × X is viewed as an arrow from y to x and composition is defined by the rule (x, y) · (y,z) = (x, z) . Definition B .4. Given a continuous map φ : U → X, the relative pair groupoid Pair (φ) is defined to be the topological groupoid whose arrow space is the fibered product U ×X U and object space is U, where an element

(x, y) ∈ U ×X U ⊂ U × U 44 David Carchedi is viewed as an arrow from y to x and composition is defined by the rule (x, y) · (y,z) = (x, z) . The pair groupoid of a space X is the relative pair groupoid of the unique map from X to the one-point space. Definition B .5. Given a topological groupoid G and a continuous map f : X → ∗ G0, there is an induced groupoid f (G), which is a topological groupoid whose object space is X such that arrows between x and y in f ∗ (G) are in bijection with arrows between f(x) and f(y) in G. When X = Uα with U = (Uα ֒→ X)α an α ∗ open cover of G0 and X → G0 the canonical map,`f (G) is denoted by GU . If in addition to this, G = (T )id for some topological space T , then this is called the ˇ Cech groupoid associated to the cover U of T and is denoted by TU . Remark. If the open cover U is instead a cover for a different Grothendieck topology on TOP, the above still makes sense. This will be important later. Remark. A Cechˇ groupoid is just a pair groupoid, for a map of the form → X. α B .2. Principal Bundles. Principal bundles for topological groups, and more` generally for topological groupoids, are classical objects of study. However, principal bundles (and many other objects involving a local triviality condition) should not be thought of as objects associated to the category TOP, but rather as objects associated to the Grothendieck site (TOP, O), where O is the open cover topology on TOP. This Grothendieck topology is defined by declaring a family of maps

(Oα → X)α to be a covering family if and only if it constitutes an open cover of X. The concept of principal bundles generalizes to other Grothendieck topologies on TOP and we will need this generality later when we introduce the compactly generated Grothendieck topology. For the remainder of this subsection, let J be an arbitrary subcanonical Grothendieck topology on TOP. Deﬁnition B .6. Given a topological groupoid G, a (left) G-space is a space E equipped with a moment map µ : E → G0 and an action map ρ : G1 ×G0 E → E, where

G1 ×G0 E / E

s G1 / G0 is the fibered product, such that the following conditions hold: i) (gh) · e = g · (h · e) whenever e is an element of E and g and h elements of G1 with domains such that the composition makes sense ii) 1µ(e) · e = e for all e ∈ E iii) µ (g · e)= t (g) for all g ∈ G1 and e ∈ E. Definition B .7. Suppose that G E is a G-space. Then the action groupoid ⋉ G E is defined to be the topological groupoid whose arrow space is G×G0 E and whose object space is E. An element

(g,e) ∈G×G0 E ⊂G× E is viewed as an arrow from e to g · e. Composition is deﬁned by the rule Compactly Generated Stacks 45

(g,e) · (h,g · e) = (hg,e) . Definition B .8. Given a topological groupoid G, the translation groupoid EG is defined to be the action groupoid G ⋉ G1 with respect to the action of G on G1 by multiplication. Definition B .9. A (left) G-bundle over a space X (with respect to J) is a (left) G-space P equipped with G-invariant projection map π : P → X which admits local sections with respect to the Grothendieck topology J. This last condition means that there exists a covering family U = (Ui → X)i in J and morphisms σi : Ui → P called local sections such that the following diagram commutes for all i:

P ⑦> ❆❆ σi ⑦⑦ ❆❆ f ⑦⑦ ❆❆ ⑦ ❆❆ ⑦⑦ Ui / X.

This condition is equivalent to requiring that the projection map is a J-cover. Such a G-bundle is called (J)-principal if the induced map,

G1 ×G0 P → P ×X P is a homeomorphism. We typically denote such a principal bundle as

G1 P ⑦⑦ µ ⑦⑦ ⑦⑦ π ⑦⑦ G0 X. Remark. To ease terminology, for the rest of this section, the term principal bundle, will refer to a J-principal bundle for our ﬁxed topology J. Deﬁnition B .10. Any topological groupoid G determines a principal G-bundle over G0 by

G1 G1 ⑧ t ⑧⑧ ⑧⑧ s ⑧⑧ G0 G0 called the unit bundle, 1G. Deﬁnition B .11. Given P and P ′, two principal G-bundle over X, a continuous map f : P → P ′ is a map of principal bundles if it respects the projection maps and is G-equivariant. It is easy to check that any such map must be an isomorphism of principal bundles. 46 David Carchedi

Definition B .12. Let f : Y → X be a map and suppose that P is a principal G-bundle over X. Then we can give Y ×X P → Y the structure of a principal G-bundle f ∗ (P ) over Y , called the pull-back bundle, in the obvious way. Definition B .13. If G is topological groupoid and P is a principal G-bundle over X, then we define the gauge groupoid Gauge (P ) to be the following topological groupoid: The fibered product

P ×G0 P / P

µ P / G0 admits a left-G-action with moment mapµ ˜ ((p, q)) = µ (p) via

g · (p, q) = (g · p,g · q) . P × P The arrow space of Gauge (P ) is the quotient G0 /G and the object space is X. An equivalence class [(p, q)] is viewed as an arrow from π (p) to π (q) (which is well deﬁned as π is G-invariant.) Composition is determined by the rule

[(p, q)] · [(q′, r)] = [(p,g · r)] ′ where g is the unique element of G1 such that g · q = q. Deﬁnition B .14. Let G and H be topological groupoids. A (left) principal G-bundle over H is a (left) principal G-bundle

G1 P ⑦⑦ µ ⑦⑦ ⑦⑦ ν ~⑦⑦ G0 H0 over H0, such that P also has the structure of a right H-bundle with moment map ν, with the G and H actions commuting in the obvious sense. We typically denote such a bundle by

G1 P ❅ H1 ❅❅ µ ❅❅ν ❅❅ ❅ G0 H0. Remark. To a continuous functor ϕ : H → G, one can canonically associate a principal G-bundle over H. It is deﬁned by putting the obvious (right) H-bundle ∗ structure on the total space of the pullback bundle ϕ0 (1G)= G1 ×G0 H0. B .3. Bicategories of Topological Groupoids. Deﬁnition B .15. A continuous functor ϕ : H → G between two topological groupoids is a J-Morita equivalence if the following two properties hold: Compactly Generated Stacks 47

i) (Essentially Surjective)

The map t ◦ pr1 : G1 ×G0 H0 → G0 admits local sections with respect to

the topology J, where G1 ×G0 H0 is the ﬁbered product

pr2 G1 ×G0 H0 / H0

pr1 ϕ

s G1 / G0. i) (Fully Faithful) The following is a ﬁbered product

ϕ H1 / G1

hs,ti hs,ti ϕ×ϕ H0 × H0 / G0 × G0.

Remark. If U is a J-cover of the object space G0 of a topological groupoid G, then the induced map GU → G is a J-Morita equivalence. Remark. We will again suppress the reference to the Grothendieck topology J; for the rest of the section a Morita equivalence will implicitly mean a J-Morita equivalence for our ﬁxed topology J. A Morita equivalence with respect to the open cover topology will be called an ordinary Mortia equivalence. Remark. The property of being a Morita equivalence is weaker than being an equivalence in the bicategory TOPGpd. In fact, a Morita equivalence is an equivalence in TOPGpd if and only if t ◦ pr1 admits a global section. However, any Morita equivalence does induce an equivalence in the bicategory Gpd after applying the forgetful 2-functor. Morita equivalences are sometimes referred to as weak equivalences.

We denote by WJ the class of Morita equivalences. The class WJ admits a right TOP −1 calculus of fractions in the sense of [23]. There is a bicategory Gpd WJ obtained from the bicategory TOPGpd by formally inverting the Morita equivalences. A 1-morphism from H to G in this bicategory is a diagram of continuous functors

K ⑦ ❄❄ w ⑦⑦ ❄❄ ⑦⑦ ❄❄ ⑦⑦ ❄❄ ⑦ H G such that w is a Morita equivalence. Definition B .16. Such a diagram is called a generalized homomorphism. There is also a well defined notion of a 2-morphism. For details see [23]. There is in fact another bicategory which one can construct from the bicategory TOPGpd. This bicategory, denoted by BunJ TOPGpd again has the same objects as TOPGpd. A 1-morphism between two topological groupoids H and G is a left principal G-bundle over H (with respect to J). There is a well defined way of composing principal bundles, for details see [15] or [17]. A 2-morphism between two such principal bundles is a biequivariant map (a map which is equivariant with 48 David Carchedi respect to the left G-action and the right H-action). A principal G-bundle over H is an equivalence in this bicategory if and only if the underlying H-bundle is also principal. By the remark following definition B .14, there is a canonical inclusion TOPGpd ֒→ BunJ TOPGpd which sends Morita equivalences to equivalences. Therefore, there is an induced map TOP −1 J TOP Gpd WJ → Bun Gpd of bicategories. Theorem B .1. The induced map TOP −1 J TOP Gpd WJ → Bun Gpd is an equivalence of bicategories. This theorem is well known. A 1-categorical version of this theorem is proven in [15], Section 2.6. The general result follows easily from [23], Section 3.4.

op B .4. Topological Stacks. Consider the bicategory GpdTOP of weak presheafs over TOP 8, that is contravariant (possibly weak) 2-functors from the category TOP into the 2-categeory of (essentially small) groupoids Gpd. Let G be a topological group. A standard example would be the weak presheaf that assigns to each space the category of principal G-bundles over that space (this category is a groupoid). More generally, let G be a topological groupoid. Then G determines a weak presheaf on TOP by the rule (id) X 7→ HomTOPGpd (X) , G . op This deﬁnes an extended Yoneda 2-functor y ˜ : TOP Gpd → GpdTOP and we have the obvious commutative diagram

y TOPop TOP / Set

( · )(id) (· )(id)

op TOPGpd / GpdTOP . y˜ Remark. y˜ preserves all weak limits. TOP Given a subcanonical Grothendieck topology J on , we denote by [G]J the associated stack on (TOP,J), aJ ◦ y˜(G), where aJ is the stackiﬁcation 2 functor (Deﬁnition A .3). It can be checked that

[G] (X) ≃ holim HomTOPGpd (XU , G) , J −−−−−−−→ U∈Cov(X) where the weak 2-colimit above is taken over a suitable bicategory of J-covers. For details in the case J is the open cover topology, see [7].

8Technically, this is not well deﬁned due to set-theoretic issues, however, this can be overcame by careful use of Grothendieck universes. We will not dwell on this and all such similar size issues in this paper. Compactly Generated Stacks 49

Remark. Since aJ preserves ﬁnite weak limits, it follows that [·]J does as well. Deﬁnition B .17. A stack X on (TOP,J) is presentable if it is equivalent to

[G]J for some topological groupoid G. In this case, G is said to be a presentation of X . We denote the full sub-bicategory of StJ (TOP) consisting of presentable stacks by PresStJ (TOP). Deﬁnition B .18. A topological stack is a presentable stack for the open cover topology on TOP. We shall denote the topological stack associated to a topological groupoid G by [G]. We shall also denote the bicategory of topological stacks by TSt. Theorem B .2. The 2-functor

aJ ◦ y˜ : TOPGpd → PresStJ (TOP) induces an equivalence of bicategories

TOP −1 ∼ TOP PJ : Gpd WJ −→ PresStJ ( ) . This theorem is well known. For example, see [23] for the case of étale topological groupoids and topological stacks with an étale atlas, and also for étale Lie groupoids and étale differentiable stacks. The preprint [3] contains much of the necessary ingredients for the proof of the case of general Lie groupoids and differentiable stacks in its so called Dictionary Lemmas. Similar statements in the case of algebraic stacks can be found in [18]. The general theorem follows again from an easy application of [23], Section 3.4. TOP TOP −1 J TOP Hence all three bicategories, PresStJ ( ), Gpd WJ and Bun Gpd are equivalent. Corollary B .1. If G is a topological groupoid, then the associated presentable stack

[G]J (is equivalent to one that) assigns to each topological space T , the groupoid of J principal G-bundles over T , BunG (T ). Deﬁnition B .19. [12] A morphism ϕ : X → Y between weak presheaves is said Y to be a J-covering morphism if for every space X and every object x ∈ (X)0, there exists a J-covering family U = (fi : Ui → X)i of X such that for each i there exists an object xi ∈ X (Ui)0 and a(n) (iso)morphism

αi : ϕ (X (fi) (xi)) → Y (fi) (x) . If X and Y are both J-stacks, J-covering morphisms are referred to as J- epimorphisms[12],[19]. There is a more intrinsic description of presentable stacks: Deﬁnition B .20. A J-atlas for a stack X over (TOP,J) is a representable (see Deﬁnition A .2) J-epimorphism p : X → X from a topological space X. Proposition B .1. [19] A stack X is presentable if and only if it has an atlas. 50 David Carchedi

Suppose that p : X → X is an atlas. Consider the weak ﬁbered product

X ×X X / X

p p X / X . Then X ×X X ⇒ X has the structure of a topological groupoid whose associated stack is X . Conversely, given a topological groupoid G, the canonical map of groupoids id (G0) → G (which is the identity objects and u on arrows) produces a morphism p : G0 → [G]J which is a representable epimorphism. We end this section by stating a technical lemma.

Lemma B .3. The presentable stack [G]J is the weak colimit of the following diagram of representables: → → H2 → H1 → H0. where the maps are the face maps of the truncated simplicial diagram. Proof. This is the content of [19] Proposition 3.19. References

[1] Théorie des topos et cohomologie étale des schémas. Tome 3. Lecture Notes in Mathematics, Vol. 305. Springer-Verlag, Berlin, 1973. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigépar M. Artin, A. Grothendieck et J. L. Verdier. Avec la collabo- ration de P. Deligne et B. Saint-Donat. [2] Masao Aoki. Hom stacks. Manuscripta Math., 119(1):37–56, 2006. [3] Kai Behrend and Ping Xu. Differentiable stacks and gerbes, 2006. arXiv:math/0605694. [4] Weimin Chen. On a notion of maps between orbifolds. I. Function spaces. Commun. Contemp. Math., 8(5):569–620, 2006. [5] Albrecht Dold. Partitions of unity in the theory of fibrations. Ann. of Math. (2), 78:223–255, 1963. [6] Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and An- gelo Vistoli. Fundamental algebraic geometry, volume 123 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. Grothendieck’s FGA explained. [7] David Gepner and AndréHenriques. Homotopy theory of orbispaces. arXiv:math/0701916, 2007. [8] Andr Haefliger. On the space of morphisms between étale groupoids, 2007. arXiv:0707.4673. [9] Ernesto Lupercio and Bernardo Uribe. Loop groupoids, gerbes, and twisted sectors on orbifolds. In Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Con- temp. Math., pages 163–184. Amer. Math. Soc., Providence, RI, 2002. [10] Jacob Lurie. Higher topos theory, volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009. [11] Saunders Mac Lane and Ieke Moerdijk. Sheaves in geometry and logic. Universitext. Springer- Verlag, New York, 1994. A first introduction to topos theory, Corrected reprint of the 1992 edition. [12] David Metzler. Topological and smooth stacks. arXiv:math/0306176, 2003. [13] Ieke Moerdijk. Classifying toposes and foliations. Ann. Inst. Fourier (Grenoble), 41(1):189– 209, 1991. [14] Ieke Moerdijk. On the weak homotopy type of étale groupoids. In Integrable systems and foliations/Feuilletages et systèmes intégrables (Montpellier, 1995), volume 145 of Progr. Math., pages 147–156. Birkhäuser Boston, Boston, MA, 1997. [15] Ieke Moerdijk and Janez Mrˇcun. Lie groupoids, sheaves and cohomology. In Poisson geometry, deformation quantisation and group representations, volume 323 of London Math. Soc. Lecture Note Ser., pages 145–272. Cambridge Univ. Press, Cambridge, 2005. Compactly Generated Stacks 51

[16] Kiiti Morita. On the product of paracompact spaces. Proc. Japan Acad., 39:559–563, 1963. [17] Janez Mrˇcun. Stability and invariants of hilsum-skandalis maps. arXiv:math/0506484v1, 1996. [18] Niko Naumann. The stack of formal groups in stable homotopy theory. Adv. Math., 215(2):569–600, 2007. [19] Behrang Noohi. Foundations of topological stacks i. arXiv:math/0503247, 2005. [20] Behrang Noohi. Homotopy types of topological stacks. arXiv:0808.3799, 2010. [21] Behrang Noohi. Mapping stacks of topological stacks. J. Reine Angew. Math., 646:117–133, 2010. [22] Martin C. Olsson. Hom-stacks and restriction of scalars. Duke Math. J., 134(1):139–164, 2006. [23] Dorette Pronk. Etendues and stacks as bicategories of fractions. Compositio Mathematica, 102(3):243–303, 1996. [24] N. E. Steenrod. A convenient category of topological spaces. Michigan Math. J., 14:133–152, 1967.